Modeling Financial Swaps and Geophysical Data Using The

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MODELING FINANCIAL SWAPS AND GEOPHYSICAL DATA USING THE

BARNDORFF-NIELSEN AND SHEPHARD MODEL

A Dissertation Submitted to the Graduate Faculty of the North Dakota State University of Agriculture and Applied Science

Semere Kidane Habtemicael

In Partial Fulﬁllment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

Major Department: Mathematics

November 2015

Fargo, North Dakota NORTH DAKOTA STATE UNIVERSITY Graduate School

Title

MODELING FINANCIAL SWAPS AND GEOPHYSICAL DATA USING THE

BARNDORFF-NIELSEN AND SHEPHARD MODEL

Semere Kidane Habtemicael

The supervisory committee certiﬁes that this dissertation complies with North Dakota State Uni- versity’s regulations and meets the accepted standards for the degree of

DOCTOR OF PHILOSOPHY

SUPERVISORY COMMITTEE:

Indranil SenGupta

Chair

Azer Akhmedov

Nikita Barabanov

Sylvio May

Approved: November 17, 2015 Benton Duncan

Date Department Chair ABSTRACT

This dissertation uses Barndoff-Nielsen and Shephard (BN-S) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock market activity and earthquake occurrence, we observe that the distributions have high kurtosis and right skewness. Consequently, such data cannot be captured by stochastic models driven by a Wiener process. Non-Gaussian processes of Ornstein-Uhlenbeck type are one of the most significant candidates for being the building blocks of models of financial economics. These models offer the possibility of capturing important distributional deviations from Gaussianity and thus these are more practical models of dependence structures. Introduced by Barndorff-Nielsen and Shephard these processes are not only convenient to model volatility in

financial market, but have an independent interest for modeling stationary time series of various kinds. For the financial data we use BN-S models to price the arbitrage-free value of volatility, variance, covariance, and correlation swap. Such swaps are used in financial markets for volatility hedging and speculation. We use the S&P 500 and NASDAQ index for parameter estimation and numerical analysis. We show that with this model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters.

For any given time interval, the earthquake magnitude data have three main properties:

(1) magnitude is a non-negative stationary stochastic process; (2) for any finite interval of time there are only finite number of jumps; (3) the sample path of the magnitude of an earthquake consists of upward jumps (significant earthquake) and a gradual decrease (aftershocks). For such geophysical data we specifically use Gamma Ornstein Uhlenbeck processes driven by a Lévyprocess to estimate a major earthquake in a certain region in California. Rigorous regression analysis is provided, and based on that, first-passage times are computed for different sets of data. Both applications demonstrate the significance of BN-S models to phenomena that follow non-Gaussian distributions.

iii ACKNOWLEDGEMENTS

First I would like to thank ALMIGHTY GOD for giving me the audacity and sanctioning me with the conﬁdence to fulﬁll this daunting task. I am indubitably grateful to my Committee

Chairman, Dr. Indranil SenGupta, for his excellent guidance, caring, and patience; for providing me with an excellent atmosphere for completing this dissertation; and for soliciting additional resources throughout my Doctoral Program endeavor. I beneﬁtted from his provocative thoughts and discussions. He provided his perceptive comments and/or constructive feedback in near real- time. Although it is impossible to list everyone to whom I am indebted, several persons deserve special mention. My mentor Drs. Indranil SenGupta, Azer Akhmedov, Nikita Barabanov, and

Sylvio for accepting to serve and their support. Without all their kind assistance, continuous support, and insightful feedback, I would not achieve and accomplish what seems to be too far to reach in a lifetime. I would like to extend a note of thanks to Mr. Curt Doetkott, Instructor of

SAS/Lead of the Statistical Consulting Services unit, at NDSU, for his untiring assistance while writing my dissertation.

My deepest appreciation and a profound thankful note also goes to Dr. Muise Ghe- bremichael for his strenuous eﬀort, motivation, sage guidance, who took the initiative to shepherd me throughout this tortuous academic journey. Dr. Ghebremichael has a special memories in my personal life, and no words could adequately express my gratitude. My wholehearted and sincere gratitude/appreciation also goes to Dr. Semhar Michael, her husband Dr. Johannes Afework and

Mr. Noah Lebasi for their hospitality, knowledge, and wisdom have supported and enlightened me over many years of friendship. Their altruistic support have been invaluable during my doctoral program and while writing this dissertation.

A note of thanks also goes to our Secretary Ms. Milan Melanie, Department of Mathematics

NDSU, for all her friendliness, assistance and support throughout my tenure at NDSU.

I thank the faculty and graduate assistants of the Mathematics department, North Dakota state University, for their advice and reminders over the last three and half years. Special thank goes to Melissa Duchsherer, John Laumann-Belth, and Matt Warner(graduate writers center), for accepting to proofreading my dissertation. A thankfull note also goes to the graduate students of

iv department of statistics, learning statistics class was fun with you guys all. I had the apportunity to learn what seems impossible in my career.

I would like to extend special thanks to Fesseha Gebremikael, a Ph. D. student at Upper

Great Plains Transportation Institute (UGPTI) for his countless beﬁtting brotherly suggestions, encouragement, who took the liberty to review, and edit my dissertation. Discussions with Fesseha always left me empowered and invigorated, and for all those moments I am very thankful. My thanks also goes to my Fargo friends, Rufael, kefali with his family, Mahder, Tadele, and to my colleague Melisa and Cory, who assisted, advised, and supported my eﬀorts over the years. I would like also to thank friends from Grand Forks area, Gashaw, Tesfay, Dr. Chernet, Dr. Daba, Enque, and Dr. Belete.

My extraordinary gratitude is also extended to my dearest parents my father Mr. Kidane

Gebreselassie my mother Wezero Tsehaynesh Unquay have been the the primary source of motivation and support for education and career. You have educated me, you have encouraged me to follow my dreams, you have been the best parents one could wish for. I thank you from the bottom of my heart. I thank my brothers Mebrahtu (with his family), Dawit, Mehari, Robiel, Kibrom, and

Aron andmy sisters Milasu (with her family),Nazret, and Elsu for the love and continued support.

I am forever indebted to my parents for their understanding, endless patience and encouragement when it was most required.

Throughout the writing of this dissertation, I have had the love, full support and tenacity of my wife Akiberet (Akutey) and my lovely God’s gift son Nathan. They are the love of my life, the rock that I can lean on and the ones who shares my best and worst moments. I greatly appreciate each one of them, for their compassionate encouragement throughout this journey. Without them, this proud and victorious episode would not have been achievable.

v DEDICATION

I would like to dedicate this doctoral dissertation to my loving parents, brothers and sisters, and

of course, my wife and son. Thanks so much for their never-ending love, support, and spirit,

during this tedious period.

vi TABLE OF CONTENTS

ABSTRACT ...... iii

ACKNOWLEDGEMENTS ...... iv

DEDICATION ...... vi

LIST OF TABLES ...... ix

LIST OF FIGURES ...... x

1. INTRODUCTION ...... 1

1.1. Types of Volatility ...... 4

1.1.1. Historical Volatility ...... 4

1.1.2. Implied Volatility ...... 4

1.2. Recent Developments in Swaps ...... 5

2. THE HULL -WHITE MODEL FOR DERIVING SWAP ...... 10

2.1. Introduction ...... 10

2.2. Variance and Volatility Swap ...... 11

2.2.1. Variance Swap ...... 11

2.2.2. Volatility Swap ...... 12

2.3. Covariance and Correlation Swap ...... 15

2.3.1. Covariance Swap ...... 15

2.3.2. Correlation Swap ...... 18

3. BARNDORFF-NIELSEN AND SHEPHARD MODEL FOR STOCK AND VOLATILITY DYNAMICS ...... 20

4. PRICING VARIANCE AND VOLATILITY SWAP FOR BN-S MODEL ...... 30

4.1. Variance Swap ...... 30

4.2. Volatility Swap ...... 31

4.3. Closed Form Solution of Volatility Swap ...... 31

4.4. Model Fitting and Parameter Estimate ...... 39

vii 4.5. Conclusion ...... 43

5. PRICING COVARIANCE AND CORRELATION SWAP FOR TWO ASSETS WITH A STOCHASTIC VOLATILITY ...... 47

5.1. Pricing Covariance Swap ...... 47

5.2. Pricing Correlation Swap ...... 57

5.3. Model Fitting and Parameter Estimate ...... 61

5.4. Conclusion ...... 62

6. ORNSTEIN-UHLENBECK PROCESS FOR GEOPHYSICAL DATA ANALYSIS . . . . 66

6.1. Introduction ...... 66

6.2. Ornstein-Uhlenbeck Processes for Geophysical Modeling ...... 69

6.3. Computation of Characteristic Function ...... 72

6.4. Analysis of the First-Passage Time ...... 74

6.5. Statistical Analysis of the Data ...... 77

6.5.1. Geophysical (Earthquake) Data ...... 78

6.5.2. Regression Analysis and Estimation of Major Events ...... 78

6.6. Conclusion ...... 80

REFERENCES ...... 85

viii LIST OF TABLES

Table Page

4.1. Parameter estimate of Gamma distribution for variance swap ...... 40

4.2. Parameter estimate of Inverse Gaussian for variance swap ...... 40

4.3. Parameter estimate of Positive tempered stable for variance swap ...... 41

4.4. Comparing errors of diﬀerent models for variance swap ...... 41

4.5. Parameter estimates for Γ volatility swap with Theorem 4.2.2 ...... 43

4.6. Parameter estimates for IG volatility swap with Theorem 4.2.2 ...... 43

4.7. Parameter estimates for PTS volatility swap with Theorem 4.2.2 ...... 44

4.8. Comparing errors of diﬀerent models for volatility swap ...... 44

4.9. Parameter estimates for Γ volatility swap with Theorem 4.3.3 ...... 45

4.10. Parameter estimates for IG volatility swap with Theorem 4.3.3 ...... 45

4.11. Parameter estimates for PTS volatility swap with Theorem 4.3.3 ...... 46

4.12. Comparing errors of diﬀerent models for volatility swap ...... 46

5.1. Parameter estimates for Variance Gamma Covariance swap ...... 64

5.2. Parameter estimates for Inverse Gaussian covariance swap ...... 64

5.3. Parameter estimates for Positive Tempered Stable covariance swap ...... 64

6.1. Results from regression analysis ...... 84

6.2. Estimation from the regression model...... 84

ix LIST OF FIGURES

Figure Page

1.1. How does swap work...... 2

1.2. Closing Price of the S&P500 Stock Price from 2010 to 2015...... 3

4.1. Fitting of variance swap ...... 42

4.2. Fitting of volatility swap ...... 45

4.3. Fitting of volatility swap with improved model ...... 46

4.4. (Left) Reliability of ﬁtting with truncated inﬁnite series for volatility swap. (Right) Zoomed in version of the left picture...... 46

5.1. Fitting of covariance swap ...... 64

5.2. (Left) Reliability of ﬁtting with truncated inﬁnite series for covariance swap. (Right) Zoomed in version of the left picture...... 65

6.1. Images comparing the raw signal and the variability estimates for two locations. The Red Hills variability is trailing the Donna Lee variability by about 1.3 seconds . . . . . 67

6.2. Graphical example of exit time...... 75

6.3. Fitting for Latitude 40.37◦ ± 0.1◦ and Longitude −124.32◦ ± 0.2◦...... 82

6.4. Fitting for Latitude −23.34◦ ± 0.5◦ and Longitude −70.30◦ ± 0.5◦...... 82

6.5. Fitting for Latitude 63.52◦ ± 0.17◦ and Longitude −147.44◦ ± 0.34◦...... 83

6.6. Fitting for Latitude −17.66◦ ± 0.03◦ and Longitude −178.76◦ ± 0.06◦...... 83

6.7. Fitting for Latitude −19.93◦ ± 0.05◦ and Longitude −178.18◦ ± 0.1◦...... 84

x 1. INTRODUCTION

A financial security is a financial contract whose value at expiration is determined by the price process of the underlying assets. A derivative is a financial security whose price depends upon or is derived from one or more underlying assets. There are two types of derivitatives: options and forwards. An option is a contract to buy or sell a financial product at a designated date, over a fixed period of time. An option gives the buyer (the owner or holder) the right, but not an obligation, to buy or sell an underlying asset or financial instrument at a price specified in the contract, which is called strike price, at a fixed period. Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck (OU) type is discussed in detail by Barndorff-Nielsen and Shephard (BN-

S) in 1997. This research has changed how ﬁnancial data is modeled, including option pricing.

Moreover, BN-S models have an interesting feature that captures stock volatility and stationary distributions. A forward contract is similar to an option that obliges the buyer to exercise the contract at the specified period of time. A swap is a financial derivative in which two counter parties agree to exchange future cash flow, where at the begining of the contract the size of the cash flow is determined. Swap is first introduced in the 1980s, and is an agreement between two

financial sectors to exchange cash flow at one or several future dates as defined in [65]. Swap is used to hedge and speculate on stock price. For example, volatility swap gives traders an exposure to profit from the risk of increase or decrease in the volatility of the stock, or hedge against these volatility risks. The four types of swaps, in order of their quantitative importance are listed below.

• An interest rate swap allows two parties to exchange a fixed and floating cash flow on investments or loans held by either parties. The most popular type of interest rate swap is

the plain vanilla swap that allows two companies to exchange cash ﬂow based on interest rate

of the same currency: i.e., fixed versus floating interest rate on a fixed date.

• A currency swap is a contract based on two currencies. An example can be an American company that wants to expand in Europe, and a European company that wants to expand in

America. Moreover, assume the interest rate to buy a currency for domestic and international

is diﬀerent. At this point, these two companies agree to exchange the interst rate to buy

1 currency.

• A commodity swap is commonly used among companies or people that use ﬁnished products or raw materials. The commodity swap is usually used to hedge against the price of a

commodity. The most common commodity swap is observed in the oil market.

• A credit default swap deals with insurance for a third-party borrower.

Figure 1.1. How does swap work.

In Figure 1.1, we see how swap works between two companies (company A versus company

B), to exchange cash flow based on fixed and floating interest rate. It literally means that, at the end of the contract, if the volatility of the market surges, then company A is obliged to pay company B the difference between fixed and floating interest rate. Likewise, if the volatility of the market falls, then company A is going to get back the difference between fixed and floating interest rate.

There are many other types of swaps including volatility, variance, covariance, and correlation. These are forward contracts whose value is determined at the begining of the contract.

Variance, volatility, covariance, and correlation swap have been an active research area within quantitative ﬁnance since the publication of Black-Scholes (BS) equation in [10]. Researchers devote a lot of time in expanding the Black-Scholes equation for pricing call and put options of a

ﬁnancial market. One of the drawbacks of Black-Scholes formula is the assumption of normality of the stock return which commonly has a right tailed distribution. This was remedied in the late

2 1980s and early 1990s using a class of inﬁnitely divisible distributions known as Levy processes.

Levy processes have a higher kurtosis and skewness than that of the Normal distribution. Since then they have been refined to take into account different variations of the Black-Scholes model and different models of the market. The Black-Scholes model also assumes that the volatility is constant, which is unrealistic given the empirical observations of the log-return. These problems have been addressed in several models. Volatility is assumed to be a deterministic function of time

(σ = σ(t)). Volatiltiy is assumed to be a function of time and current level of the stock, which mathematically means, σ = σ(t, St) which is known as a local volatility. The volatility of the log-retun of the market can be also made more realistic by incorporating stock movement from a designated period of time, which means σ = σ(t, S(t − τ))τ∈[−θ,0], θ > 0. These and others are natural extensions of a model in which volatility is a function of stochastic process.

S&P 500 index 2000 1500 Closing price 1000

0 1000 2000 3000 4000 Time

Figure 1.2. Closing Price of the S&P500 Stock Price from 2010 to 2015.

3 From Figure 1.2, we can observe a pattern of the stock price movement. First, we can clearly see that the stock price has a mean reverting property. Second, given any ﬁnite time interval, the distribution of the stock movement is stationary and makes big jumps followed by an exponential decay. Stationary distributions driven by a subordinator have the ability of capturing the big jumps.

In this dissertation we focus on modeling realized variance, volatility, covariance and correlation of the above stock dynamics. Volatility is an important element in determining the stock movement.

The higher the volatility is the riskier the market. Volatility swap gives investors an exposure of proﬁting from the increase or decrease in the stocks movement or to hedge against these risks. Here we deﬁne two types of volatility swap and conclude this chapter by reviewing literature.

1.1. Types of Volatility

1.1.1. Historical Volatility

Historical volatility is the volatility that is calculated based on the underlying stock, as it is determined in terms of the annualized standard deviation of the stock price. It is also called statistical volatility, as it calculates how much the volatility is moving during a certain time interval.

It is the standard deviation calculated using historical price data (daily, weekly, monthly, quarterly, and yearly). The log returns over a one-year period is called the annualized volatility. Historical volatiltiy is important in comparing the volatiltiy of one stock with the volatility of other stocks.

We can mathematically represent historical volatility as

v u 1 n u X ¯ 2 σ = t (Ri − R) , (1.1) n − 1 i=1

(St)i where the log-return Ri = log , and (St)i is the closing stock price at time ti for i = (St)i−1 1, 2, . . . , n, and R¯ is the mean of the log-return. Log-return of the stock is assumed to be standard normal distribution and therefore R¯ = 0.

1.1.2. Implied Volatility

Despite the fact that future volatility is not directly observable in the market, it is possible to extract the market expectation of future volatility from the options traded on public exchange (or over the counter). Such estimation is called Implied volatility. The Implied and historical volatility have two major diﬀerences. Historical volatility is directly measured by the recent movement of the

4 price of the stock over a given time (day, weeks, or yearly). Implied volatility, on the other hand, is set by the market price of the derivative contract. BS equation gives the implied volatility of the option.

The BS frame-work is ﬁrst introduced in [10]. Since then, it has become an important model in option pricing. The BS model assumes that the stock price St follows a Geometric Brownian motion which is given by

dSt = µStdt + σStdWt, (1.2) where the diﬀusion µ is the annualized expected return on the stock, σ measures the annualized stock volatility and Wt is the standard Wiener process. In this case, we call St is driven by a Wiener process.

1.2. Recent Developments in Swaps

Recent literature on valuing volatility and variance swap is growing fast. We outline a brief overview of recent developments in this area. In [8] the authors investigate swaps written on powers of realized volatility in the stochastic BN-S models. In that paper, a formula for the realized variance is derived and the swap price dynamics is represented in terms of Laplace transforms. In [68] the author gives analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes.

An analytical approximation for the valuation of volatility swaps and analyze other options with the provided analytic estimation is given in [16]. In [29] the authors have discussed the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model. A general partial diﬀerential equation approach is utilized to determine the ﬁrst two moments of the realized variance in a continuous or discrete context. This information is used to approximate the expected realized volatility via a convexity adjustment.

A new probabilistic approach using the Heston model to study variance, volatility, covariance and correlation swaps for ﬁnancial markets is given in the work by [60]. As an application, the authors provide a numerical example using S&P60 Canada Index to price swap on the volatility.

In [61, 59] variance swaps for ﬁnancial markets with underlying asset and stochastic volatility with delay are considered. They provide some analytical closed form expressions for expectation and variance of the realized continuously sampled variance. The variance swap is evaluated with delay

5 both in a risk-neutral world and in the physical world. An upper bound for delay as a measure of risk is obtained and applications two numerical examples using S&P60 Canada Index (1998-2002) and S&P500 Index (1990-1993) are provided to price variance swaps with delay. As observed in

[61], variance swap for stochastic volatility with delay is similar (but with more parameters) to variance swaps for stochastic volatility in Heston model.

In [62] the authors present a variance drift-adjusted version of the Heston model which leads to a significant improvement of the market volatility surface fitting compared to Heston model. This model has two additional parameters compared to the Heston model and, thus, it can be implemented very easily. The main idea of the proposed model is to take into account some past history of the variance process. They used the change of time method for continuous local martingales to derive a closed formula for the approximation of the volatility swap price. In [15] the authors investigated the effect of discrete sampling and asset price jumps on fair variance and volatility swap strikes. Fair discrete volatility and variance delivery prices (strikes) are derived in different models such as the Black-Scholes model, the Heston stochastic volatility model, the Merton jump-diffusion model and the Bates and Scott stochastic volatility and jump model. The fair discrete and continuous variance and volatility strikes for these models are determined analytically using variance reduction and numerical integration techniques. It was found that the effect of discrete sampling is typically small while the effect of jumps can be significant.

In [32] a model-independent lower bound on variance swap is derived. In [23] a theory of robust pricing and hedging of a weighted variance swap is developed given market prices for a finite number of co-maturing put options. Assuming no arbitrage for the put option prices, no arbitrage bounds on the weighted variance swap is deduced along with super and sub replicating strategies that enforce them. It is shown in that paper that the market quotes for variance swaps are very close to the model-free lower bounds. The main tool that is used in [23] is Föllmer’spath wise stochastic calculus. In [17] the authors develop strategies for pricing and hedging options on realized variance and volatility. These combined strategies have nice features such as readily available inputs, comprehensive and readily computable outputs, accuracy and robustness, and easy modification to price and hedge options on implied volatility. In [25] the author used Barndorff-

Nielsen and Shephard model to value variance, volatiliy, covariance, and correlation swap. In [19] it is proved that a multiple of a log contract prices a variance swap, under arbitrary exponential L´evy

6 dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving L´evyprocess, subject to integrability conditions. The valuations in some cases admit enforcement by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying assets. In a further note [18], the work in [19] is extended with G-variation, which generalizes power variation. In [18] quadratic variation is generalized to

G-variation, and the share-weighted payoﬀ problems are solved. Also, the tools developed in [18] are used to analyze and minimize the risk in a family of hedging strategies for G-variation.

Covariance swaps are a generalization of the variance swap. Covariance and correlation swaps for ﬁnancial markets with Markov-modulated volatility are analyzed in [50]. Stochastic volatility driven by two-state continuous Markov chain are considered and numerical examples are presented for two volatility indexes, VIX and VXN, for that case. In [33] pricing of derivatives written on the discretely sampled realized variance of an underlying security is considered. Two new methods are proposed to evaluate the prices of options on the discretely sampled realized variance.

The first method is based on correcting prices of options on quadratic variation by asymptotic results and the other method is exact that uses Fourier-Laplace techniques. In [56, 57] analytical methodology is developed for pricing and hedging options on the realized variance under the Heston model augmented with jumps in asset returns and variance. Moreover they analyzed the effect of the discrete sampling is analyzed on the valuation of options on the realized variance in the Heston model. A method of mixing is proposed and accurately approximates the distribution of discrete variance in the Heston model. Semi-analytical Fourier transform methods are applied for pricing shorter-term options on the realized variance. In [27] a forward characteristic function approach is implemented to price variance and volatility swaps and options on swaps that are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process.

One of the main challenges in the research on volatility, variance or covariance swaps is to obtain a closed form pricing expression that can be accurately computed. The model should not incorporate a large number of parameters to slightly improve the existing results. On the contrary, if the major improvement over the existing results is possible with almost the same number of parameters as in the existing models, then it would be a signiﬁcant improvement. This will also demonstrate the superiority of the new model. To this end, in this dissertation we consider Non-

7 Gaussian processes driven by Lévyprocess. These processes have significant potential as building blocks for stochastic models of time series in finance. Such models are mathematically tractable and it is possible to build compelling stochastic volatility models using Ornstein-Uhlenbeck processes to represent volatility. It is also well-known that log-returns from these types of models share many common properties with familiar discrete time GARCH models. In our work we use the Barndorff-

Nielsen and Shephard model for stock and volatility dynamics and implement that to obtain the arbitrage free pricing of variance, volatility, covariance, and correlation swaps. For the model we obtain closed form expressions for the arbitrage-free pricing of variance, volatility, and covariance swap and an approximation solution for correlation swap. Moreover, we show that such expressions depend only on various cumulants of the driving Lévyprocess. This model has the same (or in some cases one more) number of parameters as the Heston model and, thus, this model can be implemented very easily. Moreover, it is shown that the error estimation for this model in fitting the fair delivery price is much less than existing models. Thus, the models and pricing formulas proposed in this dissertation are simple to compute and more accurate than similar models and hence can be efficiently used in practical applications.

Finally, we have extended the Gamma-Ornstein-Uhlenbeck process, to model and analyze geophysical data. Such non-Gaussian Ornstein–Uhlenbeck processes offer the possibility of capturing important distributional deviations from Gaussianity and make the model flexible of dependence structures. It is shown that the Gamma-Ornstein–Uhlenbeck process is a possible candidate for earthquake data modeling. Rigorous regression analysis is provided and based on that the first- passage times are computed for different sets of data. It is shown that this model may be used to estimate parameters related to some major events namely major earthquakes. A detailed introduction and literature review toward modeling major earthquakes is given in the last chapter.

The structure of this dissertation is as follows. In Chapter 2 we give brief introduction to the variance, volatility, covariance and correlation swap using the Hull-White Model when the stock price follows Geometric Brownian motion. In addition, we discuss overview of pricing procedure swap when the market and volatility dynamics are driven by Gaussian processes. In Chapter 3 the

Barndorﬀ-Nielsen and Shephard (BN-S) model for stock and volatility dynamics is introduced and derived the log cumulant function and characteristic function when the volatility is driven by BDLP or subordinator. In Chapter 4, we derived the main result of variance and volatility swap if the

8 stock dynamics follows BN-S model and also used the S&P500 index to estimate model parameters and compared our model with others such as Hull-White model and Heston model. In Chapter

5, we have extended the BN-S model to ﬁnd the covariance and correlation swap for two assets.

Finally in Chapter 6, we derived a model to estimate a major earthquake using Ornstein-Uhlenbeck

(OU) process for a certain regions in California.

9 2. THE HULL -WHITE MODEL FOR DERIVING SWAP

2.1. Introduction

In mathematical ﬁnance, the Hull-White model is used to model future interest rate. The

Hull–White model is introduced by John C. Hull and Alan White in 1990, and it is still one of the popular models in capturing interest rates. The Hull-White model extends the Vasicek and

Cox-Ingersoll-Ross (CIR) models. It has a short term mean revertion (mean reversion is a theory suggesting that prices and returns eventually move back towards the long term mean or average).

In this chapter we consider that a stock dynamic follows a Geometric Brownian motion and it is given by equation (2.1) as described below. The volatility square of the log-return of the stock dynamics follows the Hull-White model given in equation (2.2). Since the Hull-White model treats the log-return of the stock price as a standard normal distiribution, its short-term average is zero.

This model is demonstrated to value the price of variance, volatility, covariance, and correlation swap.

1 dSt = St(rdt + σtdWt ). (2.1)

1 Where r is the risk-free interest rate, Wt is the standard Wiener process, and σt is the volatility of the stock at a given time t, this is given by

2 2 2 2 dσt = κσt dt + ζσt dWt , κ < 0. (2.2)

1 2 Where κ and ζ are real numbers, and Wt and Wt are independent Wiener processes. The instantaneous variance of the log-return is found by taking the variance of (2.1) which is given by

1 2 V ar(rdt + σtdWt ) = σt dt. (2.3)

10 2.2. Variance and Volatility Swap

2.2.1. Variance Swap

Deﬁnition 2.2.1. Variance swap is a forward contract in which two counter parties exchange cash-

ﬂow on future realized price which is set at the initiation of the contract. The payoﬀ of a variance swap at maturity is given by

2 N(σR(S) − Kvar), (2.4) where N is the notional amount of the swap in dollar per annualized volatility point. The holder of the variance swap receives N dollars for every time where the stock realized variance exceeds the

2 variance delivery price. σR(S) is the realized variance is the average of the instantaneous variance which is given by Z T 2 1 2 σR(S) := σs ds, (2.5) T 0 and Kvar is the delivery or exercise price for the variance swap. Valuing a variance forward price is the same as that of other derivatives. The value of a variance swap price P at expiry is given by the expected present value of a future payoﬀ in the risk-neutral world.

−rT 2 Pvar = E(e (σR(S) − Kvar)). (2.6)

In the above expression r is the risk free interest rate and T is the exercise or expiry time.

Here we assume the notional amount to be one for convenience purposes.

To value the price of the variance swap, we need to calculate the expected value of the realized variance. Moreover, we need to solve equation (2.2) completely. Notice that the variance

2 of the market varies with the variance of the stock price and if we divide equation (2.2) by σt and integrate from 0 to t, then

Z t 2 Z t Z t dσs 2 2 = κds + ζdWt 0 σs 0 0 1 σ2 = σ2 exp (κ − ζ2)t + ζW 2 . (2.7) t 0 2 t

The exponential part of the above equation indicates a shifted Gaussian distribution. Using

11 Xt 1 Itˆo’slemma, E(e ) = exp(E(Xt)+ 2 V ar(Xt)) where Xt is a Brownian motion, we get the expected value of equation (2.7)

2 2 κt E(σt ) = σ0e . (2.8)

The expected value of the realized variance is

Z T 2 1 2 E(σR(S)) = E(σs )ds T 0 Z T 1 2 κt = σ0 e dt T 0 σ2 = 0 eκT − 1 . (2.9) κT

Theorem 2.2.2. The arbitrage free price of the variance swap for the Hull-White Model is given by σ2 P = e−rT 0 eκT − 1 − K . (2.10) var κT var

2.2.2. Volatility Swap

Deﬁnition 2.2.3. Volatility swap is a forward contract on the future realized volatility of a given underlying asset.

Volatility is a statistical term which is a standard deviation of the stock return. From basic statistics deﬁnition we know that volatility is square root of variance, hence the realized volatility is given by s Z T 1 2 σR(S) := σs ds. (2.11) T 0

Volatility is an important element in determining whether one stock is risky or not. A high volatility implies the security is risky and we may not want to invest in such security. On the other hand, if the volatility is low, this means the stock is less risky, so it is good to invest in such security.

Volatility swap allows investors to trade the volatility of an asset directly, as much as they would trade in price index. The payoﬀ of a volatility swap at the trading date or maturity time T is given by

N(σR − Kvol), (2.12)

12 where N is the notional amount in dollar per annualized volatility point, σR is the realized volatility of the stock which is given by equation (2.11), and Kvol is the annualized volatility delivery price. The holder of the volatility swap at expiration receives N dollars for every point by which the stock’s realized volatility σR has exceeded the exercise price Kvol. The price of the volatility swap is the expected value of the present payoﬀ in the risk neutral-world and is given by

−rt Pvol = E e (σR − KVol) , (2.13) where r is the risk free interest rate and E(.) is the expectation with respect to some risk-neutral measure. To ﬁnd the price of the volatility swap in a risk neutral world, we need to ﬁnd the expected value of the realized volatility σR as it is the only random in equation (2.13). We observe

q 2 E(σR) = E( σR). (2.14)

It is not usually easy to ﬁnd the expected value of a square root function. However, using second degree Taylor series approximation around its mean is given by

q q Var(σ2 ) E( σ2 ) ≈ E(σ2 ) − R . (2.15) R R 2 3/2 8(E(σR) Basic statistical deﬁnition of variance gives

2 2 2 2 2 Var(σR) = E((σR) ) − (E(σR)) . (2.16)

To evaluate equation (2.15) for Hull-White model, it remains to ﬁnd the expected value of

2 2 (σR) , which is given by the lemma below.

Lemma 2.2.4. For any given time s , t and given variance of the stock by equation (2.7),

2 2 2 2 2 E(σt σs ) = (σ0) exp(κ(s + t) + ζ (t ∧ s)), (2.17) where t ∧ s = min(t, s).

13 Proof. Without loss of generality we can assume that s < t. Then using equation (2.7) we have

1 h 2 2 i E(σ2σ2) = (σ2)2exp (κ − ζ2)(s + t) E eζ(Wt +Ws ) t s 0 2 1 h 2 2 i h 2 i = (σ2)2exp (κ − ζ2)(s + t) E eζ(Wt −Ws ) E e2ζWs 0 2

2 2 2 = (σ0) exp(κ(s + t) + ζ s). (2.18)

Hence the lemma is proved .

Lemma 2.2.5. The expected value of the square of the realized variance is given by

" 2 # 2(σ2)2 1 − eκT 1 − e(2κ+ζ )T E (σ2 )2 = 0 − . (2.19) R T 2(κ + ζ2) κ 2κ + ζ2

Proof. We know that Z T 2 1 2 σR = σt dt, (2.20) T 0 which gives Z T Z T 2 2 1 2 2 (σR) = 2 σt σs dsdt. (2.21) T 0 0

Using Lemma (2.2.4) and the fact that the double integral is invariant while the variables s and t are interchangeable, we obtain

Z T Z T 2 2 1 2 2 E (σR) = E 2 σt σs dsdt T 0 0 Z T Z T 1 2 2 = 2 E(σt σs )dsdt T 0 0 2 2 Z T Z t 2(σ0) 2 = 2 exp(κ(s + t) + ζ s)dsdt T 0 s=0 " 2 # 2(σ2)2 1 − eκT 1 − e(2κ+ζ )T = 0 − , (2.22) T 2(κ + ζ2) κ 2κ + ζ2 which gives as desired.

Using equation (2.16) it can be easily shown that the realized variance of the underlying

14 asset is given by

2 ! 2(σ2)2 1 − eκT 1 − e(2κ−ζ )T σ2 2 Var(σ2 ) = 0 − − 0 (eκT − 1) . (2.23) R T 2(κ + ζ2) κ 2κ + ζ2 κT

Theorem 2.2.6. The arbitrage free price of volatility swap for the stock dynamics (2.1) and volatility dynamics (2.2) is given by

r 2 2 ! ! −rT σ0 κT Var(σR) Pvol ≈ e (e − 1) − 2 − KVol , (2.24) κT σ0 κT 3/2 8( κT (e − 1))

2 where Var(σR) can be obtained from (2.23).

2.3. Covariance and Correlation Swap

Covariance and correlation swaps are among the recent financial derivatives used to hedge and speculate using two different financial underlying assets. For example, options dependent on the movement of exchange rate, such as those who pays different currency other than the underlying currency. Such exposure to currency swaps lead to a correlation between the assets and exchange rate. This risk can be eliminated by using a covariance swap. Covariance (Correlation) swap is a forward contract on the stocks realized covariance (correlation) respectively. Covariance

(correlation) swap pays the diﬀerence between an implied covariance (correlation) respectively and the realized pairwise covariance (correlation) stock prices.

2.3.1. Covariance Swap

Deﬁnition 2.3.1. A covariance swap is a forward contract on the underlying stocks S1 and S2 in which the payoﬀ at the maturity is given by the formula

1 2 N(covR(S ,S ) − Kcov). (2.25)

The value of the covariance forward swap price P on a future realized covariance with a strike price Kcov is the expected value of the future payoﬀ in the risk-neutral world, is given by

−rT 1 2 P = E{e (covR(S ,S ) − Kcov)}, (2.26)

where r is the risk free interest rate, Kcov is the strike or exercise price of covariance swap and

15 1 2 covR(S ,S ) is the realized covariance of the two stock prices. From the above equation all other

1 2 terms are constant except covR(S ,S ) which can be calculated using the deﬁnition of quadratic variation. If the stocks dynamics of the two assets follow an exponential Brownian motion

i i i i 1 i dSt = St(r dt + σtd(Wt ) ) i = 1, 2, (2.27)

i where σt is the volatility of the log-return which follows the Hull-White model as

2 1 1 2 1 1 2 1 1 d(σt ) = κ (σt ) dt + (ζ) (σt ) dWt, κ < 0, (2.28) and 2 2 2 2 2 2 2 2 ˆ 2 d(σt ) = κ (σt ) dt + (ζ) (σt ) dWt, κ < 0. (2.29) and the two driving Wiener process are related by

2 p 2 dWˆ t = ρ dWt + 1 − ρ dW˜ t, (2.30)

i th where r is the ﬁxed interest rate of the i stock, and ζ is constant real number and Wˆ t, Wt and

W˜ t are related by the above equation (2.30), and Wt and W˜ t are independent standard Wiener processes. Then one can calculate the realized covariance as

1 cov (S1 ,S2 ) = ln S1 , ln S2 R T T T T T Z T 1 1 2 = σt σt dt. (2.31) T 0

The square bracket [, ] is the quadratic covariation of the two stocks and the solution of the above model is given by equation (2.28) is

1 (σ2)1 = (σ2)1 exp (κ1 − (ζ2)1)t + ζ1W . (2.32) t 0 2 t

The solution of equation (2.29) and using (2.30) which is given by

1 p (σ2)2 = (σ2)2 exp (κ2 − (ζ2)2)t + (ζ2)(ρ2W + 1 − ρ2W˜ )) . (2.33) t 0 2 t t

16 Now if we multiply the two above volatility square models we get

2 1 2 2 2 1 2 2 φ(t) (σt ) (σt ) = (σ0) (σ0) e , (2.34) where 1 p φ(t) = (κ1 + κ2)t − ((ζ2)1 + (ζ2)2)t + (ζ1 + ζ2ρ2)W + ζ2 1 − ρ2W˜ , (2.35) 2 t t and φ(t) is a shifted Brownian motion and remember that Wt and W˜ t are independent, which makes it easier to calculate the mean and variance of φ(t) which is given by

1 E(φ(t)) = (κ1 + κ2)t − (ζ1 + ζ2)t, (2.36) 2 and

V ar(φ(t)) = (ζ1 + ζ2ρ2)2 + (ζ2(1 − ρ2))2 t. (2.37)

Applying Itˆo’slemma and let

1 g(κ, ζ) = (κ1 + κ2) − (ζ1 + ζ2) (2.38) 2 and

h(ζ, ρ) = (ζ1 + ζ2ρ2)2 + (ζ2(1 − ρ2))2 , (2.39) the expected value of equation (2.35) is

1 E((σ2)1(σ2)2) = (σ2)1(σ2)2 exp E(φ(t)) + V ar(φ(t)) t t 0 0 2 1 = (σ2)1(σ2)2 exp g(κ, ζ)t + h(ζ, ρ)t . (2.40) 0 0 2

Hence the expected covariance is given by

2 1 2 2 Z T 1 2 (σ0) (σ0) 1 E(covR(ST ,ST )) = exp g(κ, ζ)t + h(ζ, ρ)t dt T 0 2 (σ2)1(σ2)2 = 0 0 e(g(κ,ζ)+.5h(ζ,ρ))T − 1 . (2.41) T (g(κ, ζ) + 0.5h(ζ, ρ))

17 Theorem 2.3.2. The arbitrage free covariance swap is given by

(σ2)1(σ2)2 P = e−rT 0 0 e(g(κ,ζ)+.5h(ζ,ρ))T − 1 − K . (2.42) cov T (g(κ, ζ) + 0.5h(ζ, ρ)) cov

2.3.2. Correlation Swap

Deﬁnition 2.3.3. Correlation swap is a correlation forward contract of the underlying rates S1 and S2 which payoﬀ at the maturity equal to

1 2 N(corrR(S ,S ) − KCorr). (2.43)

1 2 Where Kcorr is the strike or exercise price, N is the notional amount, corrR(S ,S ) is a correlation between two assets S1 and S2.

The correlations of the two asset from the basic statistics formula is given by

1 2 1 2 covR(S ,S ) corrR(S ,S ) = . (2.44) p 1 p 2 (VarR(S )) (VarR(S ))

To value the correlation swap in the risk neutral world we need to ﬁnd

−rT 1 2 P = e (E(corrR(S ,S )) − KCorr). (2.45)

To ﬁnd the expected value of a realized correlation is a little bit challenging, here we are going to use an approximation of squre roots which is given by ([59],p. 200) and we give the result without proof, 1 2 1 2 E(covR(S ,S )) E(corrR(S ,S )) ≈ q q . (2.46) 2 1 2 2 E((σR) ) (E(σR) )

Now since the denominator of the above equation is deterministic it can be factored to ﬁnd the expected value of the realized correlation. We will close this chapter by giving the ﬁnal theorem of the correlation swap

18 Theorem 2.3.4. The arbitrage free correlation swap price in the risk-neutral world is given by

" 2 1 2 2 # −rT (σ0) (σ0) (g(κ,ζ)+.5h(ζ,ρ))T Pcorr ≈ e e − 1 − Kcorr . p 2 1 κ1T p 2 2 κ2T 2 T (g(κ, ζ) + 0.5h(ζ, ρ)) (σ0) (e − 1)/κT (σ0) (e − 1)/κ T (2.47)

19 3. BARNDORFF-NIELSEN AND SHEPHARD MODEL FOR

STOCK AND VOLATILITY DYNAMICS

Consider a financial market without a transaction costs where a risk free asset with constant return rate r and a stock are traded up to a fixed exercise date T . Barndorff-Nielsen and Shephard

(see [6, 5]) assumed that the price process of the stock S = (St)t≥0 is deﬁned on some ﬁltered probability space (Ω, F, (Ft)0≤t≤T ,P ) and is given by:

St = S0 exp(Xt), (3.1)

2 dXt = (µ + βσt ) dt + σt dWt + ρ dZλt, (3.2)

2 2 2 dσt = −λσt dt + dZλt, σ0 > 0, (3.3) where the constants µ, β, ρ, λ ∈ R with λ > 0 and ρ ≤ 0 is the leverage eﬀect. W = (Wt) is a

Brownian motion and the process Z = (Zλt) is a subordinator ( subordinator is a real-valued Lévy process with no Gaussian component and non decreasing sample paths). Poisson process, Variance gamma and inverse Gaussian are some examples of a subordinator Lévyprocess. Barndorff-Nielsen and Shephard refer to Z as the background driving Lévyprocess (BDLP). Also W and Z are assumed to be independent and (Ft) is assumed to be the usual augmentation of the filtration generated by the pair (W, Z). This model is known in literature as Barndorff-Nielsen and Shephard model (BN-S model) and in this dissertation is referred as classical BN-S model. The log-return of the stock dynamic process of equation (3.2) is a linear process as it appeared to be a linear combination of a Brownian motion and the Lévyprocess. Also, the negative sign appearing in (3.3) makes the associated process mean-reverting. We want to mention that equation(3.3) is a non-Gaussian process as it is driven by Z (instead of W ).

Non-Gaussian processes of OU type have considerable potential as building blocks for stochastic models of observational series from a wide range of ﬁelds. They oﬀer the possibility

20 of capturing important distributional deviations from Gaussianity and for flexible modeling of dependence structures. It is been well studied that financial time series of different assets has many common features such as heavy tailed distribution and log-return, aggregational Gaussianity, quasi long range dependence. Such properties of the stock dynamics are successfully modeled by

Ornstein-Uhlenbeck(OU) type stationary stochastic process driven by a subordinators.

This model is first introduced by Barndorff Nielsen and Shephard. Since then, it becomes one of an important model in generalizing the BS model. For further reading about the BN-S model we refer the reader to existing literature (see [3, 6, 5]). Since the BN-S model involves a new idea in option pricing, it can be used in Ecomonetric analysis of realised variance and estimating stochastic volatility model. This model has been used in different literature such as

[42, 52, 53, 36, 9]. Moreover, BN-S model and its generalized version are also used in pricing exotic options (see [54, 55, 21]).

The formal deﬁnition of L´evyprocess is given below.

Definition 3.0.5. ([63],p68) Lévyprocess: A cadlag stochastic process (Xt)t≥0 on (Ω, F, P) with values in R such that X0 = 0 is called a Lévyprocess if it possesses the following properties:

• Independent Increment: for every increasing sequence of times t0, t1, . . . , tn, the random vari-

ables Xt0 ,Xt1 − Xt0 ,...,Xtn − Xtn−1 are independent.

• Stationary increaments: the law of Xt+h − Xt does not depend on t.

• Stochastic Continuity: ∀ > 0, limh→0 P(|Xt+h − Xt| ≥ ) = 0.

A large family of mean reverting processes can be constructed using a Lévyprocess as a driving noise. Positiveness and the choice of marginal distribution can be urged on those Lévy process. These Lévy-driven processes are known as non-Gaussian Ornstein-Uhlenbeck processes or simply Ornstein-Uhlenbeck processes. One of the most significant candidate for being the building block of financial economics is Non-Gaussian processes of OU. The deviation from Gaussianity can be captured by those models.

As it is well established by Barndorff-Nielsen and Shephard (see [6, 5]), these processes are not only appropriate to model volatility in financial market, but have also an independent interest for modeling stationary time series of different kinds. In this Chapter, we define properties

21 of stationary disrtibution driven by L´evyprocesses and Show how to solve them. Log-laplace transform and characteristic function of a stationary process is also derived.

We assume that the BDLP Z satisﬁes the assumptions as described in [42]. The assumptions are as follows.

Assumption 1. Z has no deterministic drift and its L´evymeasures have densities w(x) . Thus by [51] (Theorem 19.3) the cumulant transform

κ(θ) = log E[eθZ1 ], (3.4) where it exists, takes the form κ(θ) = R (eθx − 1)w(x) dx. R+

Assumption 2. Letting θˆ = sup{θ ∈ R : κ(θ) < +∞}, then θˆ > 0.

Assumption 3. limθ→θˆ κ(θ) = +∞.

Then it is shown in ([42] Theorem 3.2) that there exists an equivalent martingale measure

(EMM) under which the equations (3.2) and (3.3) are transformed into the following equations.

dXt = bt dt + σt dWt + ρ dZλt (3.5)

2 2 2 dσt = −λσt dt + dZλt, σ0 > 0, (3.6) where 1 b = (r − λκ(ρ) − σ2), (3.7) t 2 t where Wt and Zλt are is Brownian motion and L´evy process respectively with respect to the equivalent martingale measure, bt is the appreciation rate, and κ(θ) is the cumulant transform for

Z1 under the new measure. The heuristic derivation of the above EMM is given in ([41], ch 6). For the rest of this section we assume that the risk-neutral dynamics of the stock price and volatility are given by (3.1), (3.5) and (3.6) and we derive the formula for the price of variance, volatility, covariance, and correlatin swap using this model.

Equation (3.6) is a linear stochastic diﬀerential equation and can be solved easly using an

22 integrating factor as it is given by

Z t 2 −λt 2 −λ(t−s) σt = e σ0 + e dZλs. (3.8) 0

2 2 Since the process σ = (σt ) is driven by a subordinator, it is strictly positive and bounded 2 2 from below by the deterministic function σ0 exp(−λt). Moreover, the stationary distribution σt jumps at the same time point of the subordinator but tailed oﬀ due to the negative sign. The instantaneous variance the log return of the stock dynamics is given by calculating the variance

2 2 of equation (3.5) which is (σt + ρ λVar[Z1]) dt . Therefore the continuous realized variance in the interval [0,T ] is the average of the instantaneous variance given by

Z T 2 1 2 2 σR = σt dt + ρ λVar[Z1]. (3.9) T 0

Lemma 3.0.6. The realized variance is given by

Z T 2 1 −1 −λT 2 −1 −λ(T −s) 2 σR = λ (1 − e )σ0 + λ 1 − e dZλs + ρ λVar[Z1]. (3.10) T 0

R λt s Proof. Substituting equation (3.8) and using the integration by parts letting U = 0 e dZs and dV = e−λtdt which gives

Z T Z λt 2 1 −λt 2 −λt s 2 σR = e σ0 + e e dZsdt dt + ρ λVar[Z1] T 0 0 Z T Z λt 1 −1 −λT 2 −λt s 2 = λ (1 − e )σ0 + e e dZs dt + ρ λVar[Z1] T 0 0 Z λt Z T 1 −1 −λT 2 −1 −λt s T 1 2 = λ (1 − e )σ0 + e e dZs|0 + dZλt + ρ λVar[Z1] T λ 0 T 0 Z T 1 −λT 2 −λ(T −s) 2 = (1 − e )σ0 + 1 − e dZλs + ρ λVar[Z1]. (3.11) λT 0

The purpose of this dissertation is to investigate the variance, volatility, covariance, and correlation swap for a BN-S type model. For that we need to ﬁnd the mean and variance of the realized stock variance. In the above lemma all the terms are deterministic except the subordinator.

23 We investiigate this in more details for the rest of this section. Below we are going to state some deﬁnitions and theorems which help us to ﬁnd the mean and varaince.

Deﬁnition 3.0.7. (Self-decomposability, [5]) A probability measure P on R is said to be self- decomposable or to belong to the L´evyclass L, if for each l > 0 there exists a probability measure

Ql on R such that −l Φ(ζ) = Φ(e ζ)Φl(ζ), (3.12)

where Φ and Φl denote the characteristic functions of P and Ql, respectively. A random variable X with law in L is also called self-decomposable.

The following two theorems give a relation between self-decomposability and L´evyprocesses.

For the proofs see [4, 67].

Theorem 3.0.8. (Stationarity, [67]) If X is self-decomposable then there exists a stationary

2 2 2 d stochastic process {σ (t)}t≥0, and a L´evyprocess {Zt}t≥0, independent of σ0, such that σt = X for all t ≥ 0 and Z t 2 2 σt = exp(−λt)σ0 + exp (−λ(t − s)) dZλs, for all λ > 0. (3.13) 0

2 Conversely, if {σt }t≥0, is a stationary stochastic process and {Zt}t≥0 is a L´evyprocess independent 2 2 of σ0, such that {σt } and {Zt} satisfy

2 2 2 dσt = −λσt dt + dZλt, σ0 > 0, (3.14)

2 for all λ > 0, then σt is self-decomposable.

Theorem 3.0.9. (Jurek and Vervaat,[31]) A random variable X has law in L if and only if

R ∞ −t X has a representation of the form X = 0 e dZt, where Zt is a Lévyprocess. In this case the R ∞ t Lévymeasure U and W of X and Z1 are related by U(dx) = 0 W (e dx) dt. In addition, if u(x), the Lévydensity of U is differentiable, then the Lévymeasure W has a density w, and u and w are related by

w(x) = −u(x) − xu0(x). (3.15)

It is clear from ([51] Theorem 17.5(ii)) that for any self-decomposable law D there exists a

24 L´evyprocess Z such that the process of OU type driven by Z has invariant distribution given by

It is well known that inverse Gaussian (IG) distributions and variance gamma distributions

2 are self-decomposable. Suppose that the stationary distribution of σt is given by IG(δ, γ) law.

IG(δ, γ) is concentrated on R+ and has probability density

1 δ2x−1 + γ2x p(x) = √ δeδγx−3/2 exp − , γ ≥ 0, δ > 0. (3.16) 2π 2

2 γ Since IG is inﬁnitely divisible, for all n ≥ 2 the sum of (σt )n with distribution as IG(δ, n ) 2 have the same distribution as of (σt ). Mathematically this is the same as

n 2 d X 2 σt = (σt )i. (3.17) i=0

2 Now it is easy to see that the L´evydensity of (σt ) as the limit of

lim nσ2 (dx) = νd(x). (3.18) n→∞ t n

Here we give the result of an Inverse Gaussian L´evydensity without proof, for detailed proof we refer to [41]. 1 u(x) = √ δx−3/2 exp(−γ2x/2), x > 0. (3.19) 2π

δ − 3 2 − 1 γ2x Applying theorem (3.0.9), Lévydensity of Z is given by w(x) = √ x 2 (1 + γ x)e 2 . 1 2 2π 2 Likewise if the stationary distribution of σt is given by gamma law Γ(ν, α) and knowing gamma density as an infinitely divisible Lévyprocess, its Lévydensity of Γ(ν, α) is given by u(x) =

−1 −αx −αx νx e , x > 0, then once again by (3.0.9) we obtain the L´evydensity of Z1 as w(x) = ναe , x > 0.

Here we deﬁne the (unconditional) cumulant generating functions or the log Laplace trans-

2 forms of σt and Z1 (if they exist) by

u u 2 κ¯ (θ) = log(E [exp(θσt )]), (3.20)

25 and

u u κ (θ) = log(E [exp(θZ1)]), (3.21) where the superscript “u” stands for unconditional. Moreover when we say unconditional which

2 means the distribution is independent of σ0. The relation of equations (3.20) and (3.21) are based on the following key result from L´evyprocesses. We state this as a lemma and the proof can be found in [41].

Lemma 3.0.10. (Key,Formula,Barndorﬀ-Nielsen and Shephard [2], p.5) Let f denote a continuous function, Z a L´evyprocess and set Y = R f(t)dZ . Then R+ t

κY (θ) = κ(Z1)(θf(t))dt. (3.22) R+

Z ∞ Using the above lemma it is easy to see the relation asκ ¯u(θ) = κu(θ exp(−s))ds. 0 This relation can be expressed using derivative as it is proved in equation (6.4) and is given by d(¯κu(θ)) κu(θ) = θ . The expected value and variance is the ﬁrst and second derivative of the cu- dθ mulant generating function evaluated at θ = 0 when they exist. It follows that if we write the

2 cumulant of Z1 and σt (when they exist) as κm andκ ¯m (m = 1, 2, 3,...) respectively where m denotes the mth derivative, then

u u κm = mκ¯m. for m = 1, 2, 3 .... (3.23)

u u 2 u u 2 Therefore, E (Z1) = E (σt ) and Var (Z1) = 2Var (σt ). However, since Z is independent 2 of σ0 this relations can also be written as

2 u u 2 E(Z1) := E(Z1|σ0) = E (Z1) = E (σt ), (3.24) and

2 u u 2 Var(Z1) := Var(Z1|σ0) = Var (Z1) = 2Var (σt ). (3.25)

In general

u κm = mκ¯m, for m = 1, 2, 3 .... (3.26)

26 2 Note that in present setting though the unconditional distribution of σt is stationary, the 2 2 conditional distribution of σt given σ0 is not stationary. However due to the independence of 2 the subordinator Z and σ0 and the relation (3.8), there is a tractable way of dealing with such conditional distribution in terms of parameters of corresponding unconditional distribution. We conclude this section with derivations of ﬁrst two cumulants for Z1 when the unconditional station-

2 ary distribution of σt is given by IG, Γ, or positive tempered stable (PTS) processes. In the next 2 section we use these results to compute the conditional cumulants of σt .

2 Lemma 3.0.11. If the stationary distribution of σt is given by an IG(δ, γ) then its log cumulant function is given by 2 p κ¯u(θ) = log(Eu(eθσt )) = δ(γ − γ2 − 2θ). (3.27)

2 Proof. Since σt follows inverse Gaussian distribution and using the deﬁnition of expected value we can deduce

2 κ¯u(θ) = log(Eu(eθσt ))

Z ∞ 2 −1 2 1 θx δγ −3/2 − δ x +γ x = log √ δe e x e 2 dx 0 2π Z ∞ √ √ 2 −1 2 1 δ γ2−2θ−δ( γ2−2θ−γ) −3/2 − δ x +(γ −2θ)x = log √ δe x e 2 dx 0 2π √ Z ∞ √ 2 −1 2 −δ( γ2−2θ−γ) 1 δ γ−2θ −3/2 − δ x +(γ −2θ)x = log e √ δe x e 2 dx 0 2π p = −δ( γ2 − 2θ − γ). (3.28)

Therefore Using the above relation or from equation (6.4) the log cumulant function of Z1 can be deduced as follows

d d h p i δθ κ(θ) = θ (¯κu(θ)) = θ δ(γ − γ2 − 2θ) = . (3.29) dθ dθ pγ2 − 2θ

Since the expected value of Z1 is the ﬁrst derivative of the log cumulant function at θ = 0

27 which is given by

d δθ −1 E(Z1) = (p ) = δγ . (3.30) dθ γ2 − 2θ θ=0

Similarly the variance of Z1 can be obtained by

! d2 δθ −3 Var(Z1) = 2 p = 2δγ . (3.31) dθ γ2 − 2θ θ=0

2 Lemma 3.0.12. If the stationary distribution of σt is given by an Γ(ν, α) law then its log cumulant function is given by

2 κ¯u(θ) = log(Eu(eθσt )) = −ν log(1 − θα−1). (3.32)

2 αν ν−1 −αx Proof. If σt follows a Γ(ν, α) with its density function is given by p(x) = Γ(ν) x e then the log cumulant function is given by

2 κ¯u(θ) = log(Eu(eθσt )) Z ∞ αν = log eθx xν−1e−αxdx 0 Γ(ν) Z ∞ ν ν α (α − θ) ν−1 −(α−θ)x = log ν x e dx 0 Γ(ν)(α − θ) α ν Z ∞ (α − θ)ν = log xν−1e−(α−θ)xdx α − θ 0 Γ(ν) α = ν log . (3.33) α − θ

Therefore using the above lemma the log cumulant function of Z1 can be also found as

d d νθ κ(θ) = θ (¯κu(θ)) = θ −ν log(1 − θα−1) = . (3.34) dθ dθ α − θ

Similarly the expected and variance of Z1 can be calculated using the log cumulant function which is given by

d νθ −1 E(Z1) = ( ) = να , (3.35) dθ α − θ θ=0

28 and

2 d νθ −2 Var(Z1) = = 2να . (3.36) dθ2 α − θ θ=0

So far we have seen a self-decomposable process having two parameters. Here we are going to add one more self-decomposable L´evyprocess which has three parameters and is called

Positive Tempered Stable (PTS). For further reading regarding PTS L´evyprocess we refer to (see

[35, 49, 66, 13, 14]). Now consider a positive stable PS(κ, δ) process whose its L´evydensity for

k κ −1−κ such a process is given by u(y) = δ2 Γ(1−κ) y . Unfortunately, the probability density function pS(y; κ, δ) of PS(κ, δ) of Positive stable process is unknown in general. However the Probability density function of PTS(κ, δ, γ) family is obtained by exponentially tilting the probability density function pS(y; κ, δ) and is given by:

1 p(y; κ, δ, γ) = eδγ exp − γ2y p (y; κ, δ), y > 0. (3.37) 2 S

This is not in general known in simple form. However, for PTS(κ, δ, γ) process and its L´evy density is given by (see [7])

κ 1 u(x) = δγ−2κ y−κ−1 exp − γ2y , y, δ > 0, 0 < κ < 1, γ ≥ 0. (3.38) Γ(κ)Γ(1 − κ) 2

2 Now again if you assume the stationary distribution of σt follows a PTS(κ, δ, γ) law. Then u 1/κ κ u 2 κ−1 u 2 κ−2 κ¯ (θ) = δγ − δ(γ − 2θ) , and E (σt ) = 2κδγ κ and Var (σt ) = 4κ(1 − κ)δγ κ . Hence in this case

κ−1 E(Z1) = 2κδγ κ , (3.39) and

κ−2 Var(Z1) = 8κ(1 − κ)δγ κ . (3.40)

Now we have seen enough background to be able to come up to our result of valuing price of the variance, volatility, covariance and correlation swap if the stock dynamics follows a BN-S model.

The rest of this dissertation discusses particularly on our main result and numerical analysis.

29 4. PRICING VARIANCE AND VOLATILITY SWAP FOR

BN-S MODEL

4.1. Variance Swap

Deﬁnition 4.1.1. A variance swap is a forward contract on the realized variance. The payoﬀ of a variance swap at expiry is given by

2 N(σR − Kvar). (4.1)

Furnished with the BN-S model and the understanding of the variance process, in this chapter we are going to prove the main result related to the arbitrage-free pricing of variance and volatility swap. Since variance swap is easy to implement, we are going to derive the price of the variance swap ﬁrst, then using equation (2.15) we showed an approximate estimate for volatility swap price. However, the Taylor series approximation may not converge if the stock market is not stable, and it might not be appropriate to use equation (2.15) when there is highly volatile market and, in this section we also derive a closed form solution for volatility swap pricing for general case.

Theorem 4.1.2. The arbitrage free price of the variance swap is given by

1 P = e−rT λ−1 1 − e−λT σ2 − κ + κ T + ρ2λκ − K , (4.2) Var T 0 1 1 2 Var

where κ1 and κ2 are the ﬁrst cumulant (i.e., the expected value) and the second cumulant (i.e., the variance) of Z1 respectively.

2 Proof. The (conditional given σ0) expected value of equation (3.10) gives the value

Z T 2 1 −1 −λT 2 −1 −λ(T −s) 2 E(σR) = λ (1 − e )σ0 + λ κ1 1 − e λ ds + ρ λVar[Z1] T 0 1 = λ−1(1 − e−λT )σ2 + κ (T − λ−1(1 − e−λT )) + ρ2λκ . (4.3) T 0 1 2

Hence the theorem follows from simpliﬁcation of (4.3).

30 4.2. Volatility Swap

Deﬁnition 4.2.1. A volatility swap is a forward contract on future realized volatility of a given underlying asset. The payoﬀ of a volatility swap at the maturity T is given by

N(σR − Kvol), (4.4)

where N is the notional amount in dollar, σR is the realized volatility and KVol is the annualized volatility delivery price.

Theorem 4.2.2. The arbitrage free value of the volatility swap is given by

r 1 P ≈ e−rT λ−1 (1 − e−λT ) σ2 − κ + κ T + ρ2λκ Vol T 0 1 1 2 λ−2 −λT 3 1 −2λT T 2 κ2(2e − 2 − 2 e + λT ) − − KVol , (4.5) 1 −1 −λT 2 2 3/2 8 T λ (1 − e ) σ0 − κ1 + κ1T + ρ λκ2 where κ2 is the second cumulant (i.e., the variance) of Z1.

2 2 Proof. The (conditional given σ0) variance of σR can be obtained from the following computation.

−1 Z T 2 λ −λ(T −s) Var(σR) = Var 1 − e dZλs T 0 −2 Z λT λ −s2 = 2 κ2 1 − e ds T 0 λ−2 3 1 = κ (2e−λT − − e−2λT + λT ). (4.6) T 2 2 2 2

2 Hence the theorem follows from (2.15) with the substitution of E(σR) from (4.3) and 2 Var(σR) from the above expression.

4.3. Closed Form Solution of Volatility Swap

Since equation (2.15) uses Taylor series expansion around the mean of the realized variance.

2 It assumes that, the long-term expected value of the realized variance, E(σR) < 1, which is not usually the case, as the market might be highly volatile. For example during the market crash 2008 where the stock price was ﬂuctuating and unpredictable. Since it is suﬃcient to know E(σR) to

ﬁnd the price of the volatility swap, in this section we ﬁnd an analytical formula for E(σR) where

31 2 the terms can be computed using the parameters of unconditional distribution of σt .

Let us assume At, be any stochastic process deﬁned on 0 ≤ t ≤ T which is independent of

2 σ0. Assume that the characteristic function and the cumulant generating function of At is given by ΦAt (θ) = E(exp(iθAt)) and κAt (θ) = log E(exp(θAt)) respectively. Since At is independent 2 of σ0 there is no any diﬀerence between superscript u and without u, for that case we omit the subscript. The relation between the characteristic function and the cumulant generating function of At is given by

ΦAt (θ) = exp[κAt (iθ)]. (4.7)

It is easy to see the above equality as

κAt (iθ) = log(E(exp(iθAt))) = log(ΦAt (θ)). (4.8)

Lemma 4.3.1. The moments of At can be obtained from ΦAt (θ) by

k k k d ΦAt (θ) E(At ) = (−i) , k = 1, 2,.... (4.9) dθk θ=0

Proof. From the deﬁnition of expected value for a continuous functions and assume P is some distribution of At we have

dk dk Z ∞ k (ΦAt (θ)) = k (exp(iθAt))d(P (At)) dθ dθ 0 Z ∞ dk = k (exp(iθAt))d(P (At) 0 dθ Z ∞ k = ((iAt) exp(iθAt))d(P (At) 0 Z ∞ k = ((iAt) d(P (At) 0 k k = i E(At ). (4.10)

The fourth step is found by substituting θ = 0.

Z λt −s Lemma 4.3.2. Suppose that At = α + (1 − e ) dZs, where α ∈ R is a constant, and 0 ≤ t ≤ T . 0

32 Then Z λt −s ΦAt (θ) = exp iθα + κ(iθ(1 − e )) ds , (4.11) 0 where κ(·) is the cumulant generating function for Z1. The moments of At are given by

k k E(At ) = (−i) gk(0), k = 1, 2,..., (4.12) where Z λt −s 0 −s g1(θ) = i α + (1 − e )κ (iθ(1 − e )) ds , (4.13) 0 and

0 gk+1(θ) = g1(θ)gk(θ) + gk(θ), k = 1, 2,.... (4.14)

In the above formulas prime represents the derivative with respect to the parameter in parenthesis.

Proof. We have

κAt (iθ) = log E(exp(iθAt)) Z λt −s = log E exp(iθα) exp(iθ (1 − e )dZs) 0 n !! X −si = iθα + log E exp iθ (1 − e )(Zsi − Zsi−1 ) i=1 n X si = iθα + log E exp(iθ(1 − e )Z(si−si−1)) i=1 Z λt −s = iθα + log E exp(iθ(1 − e )Z1) ds 0 Z λt −s = iθα + κZ1 (iθ(1 − e )) ds. (4.15) 0

Step three is using the fact that the BDLP Z has a ﬁnite variation on any closed interval and using

(4.7) we obtain (4.11).

To prove the formula for moments we observe κ(0) = 0. By diﬀerentiation (4.11) and using

k (4.9) we obtain E(At) = −ig1(0). The results related to E(At ) for k = 2, 3,... follows from induction.

33 (k) Note that κ (0) = κk, for k = 1, 2,... . Using Lemma 4.3.2 we can compute any moment

R λt −s for At = α + 0 (1 − e ) dZs in terms of cumulants of Z. For example

Z λt −s −λt E(At) = −ig1(0) = α + κ1(1 − e ) ds = α + κ1(λt − 1 + e ). (4.16) 0

2 2 2 0 E(At ) = (−i) g2(0) = −(g1(0) + g1(0)) Z λt 2 Z λt −s −s 2 = α + κ1(1 − e ) ds + κ2(1 − e ) ds 0 0 2 3 1 = α + κ (λt − 1 + e−λt) + κ 2e−λt − − e−2λt + λt . (4.17) 1 2 2 2

Since the realized variance is a ﬁnite number, it is necessary to bound by a constant real

2 2 number from above. Assume that σR < β , for some β > 0. For example, since σt is expressed in percentage, for a stable market, where it does not have “crash-like fluctuations” β = 1 is a very √ reasonable assumption. We also note that for |x| < 1, 1 + x can be represented by the convergent series ∞ √ X (−1)k+1(2k)! 1 + x = xk. (4.18) 4k(k!)2(2k − 1) k=0 The theorem which is given below gives an analytic formula for the arbitrage free value of the volatility swap. The theorem gives the arbitrage free value of the volatility swap in terms of a convergent infinite series. As the series converges very fast it is reasonable to take the first few terms for analysis purpose.

2 2 Theorem 4.3.3. Assume that σR < β , for some β > 0. Then the arbitrage free value of the volatility swap is given by

∞ ! X (−1)k+1(2k)! 1 P = e−rT E(Ak ) − K , (4.19) vol 4k(k!)2(2k − 1) β2k−1λkT k T Vol k=0

q 2 σ0 −λT 2 R λT −s 2 2 where c1 = λT (1 − e ) + ρ λκ2 and AT = α + 0 (1 − e ) dZs, with α = λT (c1 − β ). The k quantities E(AT ), k = 1, 2,... can be computed using Lemma 4.3.2.

34 Proof. We obtain from (3.10)

2 Z T 2 σ0 −λT 2 1 −λ(T −s) σR = (1 − e ) + ρ λκ2 + 1 − e dZλs λT λT 0 2 Z λT σ0 −λT 2 1 −s = (1 − e ) + ρ λκ2 + 1 − e dZs λT λT 0 2 σ0 −λT 2 1 R λT −s ! (1 − e ) + ρ λκ2 + (1 − e ) dZs = β2 λT λT 0 β2

2 2 R λT −s ! λT (c − β ) + (1 − e ) dZs = β2 1 + 1 0 β2λT

R λT −s ! α + (1 − e ) dZs = β2 1 + 0 , (4.20) β2λT

2 2 σ0 −λT 2 2 2 where c1 = λT (1 − e ) + ρ λκ2, and α = λT (c1 − β ). By the construction we have

R λT −s α + (1 − e ) dZs 0 < 1. (4.21) 2 β λT

Therefore we obtain

1/2 R λT −s ! α + (1 − e ) dZs σ = β 1 + 0 R β2λT ∞ k X (−1)k+1(2k)! 1 Z λT = α + 1 − e−s dZ . (4.22) 4k(k!)2(2k − 1) β2k−1λkT k s k=0 0

Therefore

∞ k X (−1)k+1(2k)! 1 Z λT E(σ ) = E α + 1 − e−s dZ R 4k(k!)2(2k − 1) β2k−1λkT k s k=0 0 ∞ X (−1)k+1(2k)! 1 = E(Ak ). (4.23) 4k(k!)2(2k − 1) β2k−1λkT k T k=0

It is intuitive that the inﬁnite series in (4.23) converges from (4.18) and (4.21). Thus (4.23)

k gives an analytical formula for computation of E(σR) where the quantities E(AT ), k = 1, 2,... can be computed using Lemma 4.3.2. Thus the theorem follows from the fact that the price of a

−rT volatility swap in a risk-neutral world is Pvol = e (E(σR) − KVol).

35 P∞ (−1)k+1(2k)! 1 k Next we consider the inﬁnite series k=0 4k(k!)2(2k−1) β2k−1λkT k E(AT ) of Theorem 4.3.3. 2 It is clear from the proof of that theorem that AT /β λT < 0. It is also clear from (4.21) that

2 |AT /β λT | < 1.

Theorem 4.3.4. Suppose that AT is given by Theorem 4.3.3. Then the quantity E(σR) can be approximated by n-th partial sum

n−1 X (−1)k+1(2k)! 1 E(Ak ), (4.24) 4k(k!)2(2k − 1) β2k−1λkT k T k=0 with the absolute error of approximation less than the quantity β 1√ , for n ≥ 1. (2n−1) 3n+1

Proof. The inﬁnite series representation of E(σR) is an alternating series and therefore

n−1 X (−1)k+1(2k)! 1 E(σ ) − E(Ak ) R 4k(k!)2(2k − 1) β2k−1λkT k T k=0 n β 2n AT ≤ E 4n(2n − 1) n β2λT β 2n < . (4.25) 4n(2n − 1) n

It can be proved by induction the lower and upper bound of the following inequality which is given by √ 4n n 2n 4n ≤ ≤ √ , (4.26) 2n n 3n + 1 for all n ≥ 1. Therefore we obtain

n−1 k+1 X (−1) (2k)! 1 k 1 E(σR) − E(A ) < β √ . (4.27) 4k(k!)2(2k − 1) β2k−1λkT k T (2n − 1) 3n + 1 k=0

The constant β can be used as a “control parameter” that improves the rate of convergence

P∞ (−1)k+1(2k)! 1 k of the inﬁnite series k=0 4k(k!)2(2k−1) β2k−1λkT k E(AT ). This is shown in the next theorem.

Theorem 4.3.5. Suppose that AT is given by Theorem 4.3.3 and it is possible to choose β so that

2 1 |AT /β λT | < 2+ , for some > 0. Then the quantity E(σR) can be approximated by the n-th

36 partial sum of the inﬁnite series

∞ X (−1)k+1(2k)! 1 E(Ak ), (4.28) 4k(k!)2(2k − 1) β2k−1λkT k T k=0 with the absolute error of approximation less than the quantity β 1√ 1 , for n ≥ 1, 1 (2n−1) 3n+1 (1+)n 1 1+ 2 where β1 is a constant and is equal to β 2+ .

Proof. It is easy to obtain that

n−1 k+1 k n+1 n ! X (−1) (2k)! AT (−1) (2n)! AT 1 σR = β + , (4.29) 4k(k!)2(2k − 1) β2λT 4n(n!)2(2n − 1) β2λT n− 1 k=0 (1 + µ) 2

A for some µ between T and 0. Thus the assumption |A /β2λT | < 1 gives − 1 < µ. We have β2λT T 2+ 2+

n−1 X (−1)k+1(2k)! 1 E(σ ) = E(Ak ) + E [R ] , (4.30) R 4k(k!)2(2k − 1) β2k−1λkT k T n k=0 where we denote the error term by

n+1 n (−1) (2n)! AT 1 Rn = β . (4.31) n 2 2 n− 1 4 (n!) (2n − 1) β λT (1 + µ) 2

We obtain

n β 2n AT 1 |Rn| = n 2 n− 1 4 (2n − 1) n β λT |1 + µ| 2 β 2n 1 n 1 < n 1 4 (2n − 1) n 2 + 1 n− 2 (1 − 2+ ) 1 β 2n 1 + 2 1 = . (4.32) 4n(2n − 1) n 2 + (1 + )n

Therefore using (4.26) we obtain

1 1 |Rn| < β1 √ , (4.33) (2n − 1) 3n + 1 (1 + )n

37 and consequently,

1 1 |E(Rn)| ≤ E(|Rn|) < β1 √ , (4.34) (2n − 1) 3n + 1 (1 + )n

1 1+ 2 where β1 = β 2+ .

The next theorem gives a control on estimation of regression parameters based on partial sum approximation of E(σR). For convenience we denote

n−1 X (−1)k+1(2k)! 1 S = E(Ak ). (4.35) n 4k(k!)2(2k − 1) β2k−1λkT k T k=0

We note that Sn depends on λ and various parameters of the stochastic process AT . To emphasis this dependence, in the next theorem, we write Sn as Sn(p), where p stands for all parameters that govern Sn. Also, the value of E(σR) computed based on the set of parameter p is denoted as E(σR, p).

Theorem 4.3.6. Let DT be a ﬁnite set of empirical data for volatility delivery prices with various maturity days T such that 0 < T ≤ Tmax, for some Tmax > 0. Suppose for > 0, there

(n) (n) exists a set of parameters p such that max0 0. If er-

(k) (n1) rors max0

Proof. It is clear that for n1, n2 ≥ n,

(n1) (n2) max |Sn1 (p ) − Sn2 (p )| 0

(n1) (n2) < max |DT − Sn1 (p )| + max |DT − Sn2 (p )| 0

By Theorem 4.3.4 it is clear that for for a given > 0, suﬃciently large n1 and n2 can be chosen such that (n1) (n1) max |E(σR, p ) − Sn1 (p )| < , (4.37) 0

38 and (n2) (n2) max |E(σR, p ) − Sn2 (p )| < . (4.38) 0

(n1) (n2) Hence the result max0

4.4. Model Fitting and Parameter Estimate

In this chapter we demonstrate the theoretical result using numerical data of the stock price. We also show that the performances of our results agree with the empirical data better (with respect to various measures of goodness of ﬁt as described below), than the existing comparable models. We use Theorems 4.1.2, 4.2.2 and 4.3.3 to calibrate fair delivery price. We use closing stock prices of the S&P 500 index for 943 trading dates from 12/05/2011 to 09/04/2015. Once calibration is performed over the described historical data set, we obtain the model parameters that can be used to price the ﬁxed leg (fair delivery price) of the variance or volatility swap. For goodness of

ﬁt of the calibration of fair delivery price, we use the absolute percentage error (APE), the average absolute error (AAE), the average relative percentage error (ARPE) and the root-mean-square error (RMSE) given by the following formulas.

1 X |market price − model price| AP E = , (4.39) mean price data points data points

X |market price − model price| AAE = , (4.40) data points data points

1 X |market price − model price| ARP E = , (4.41) data points data points data points

v u X (market price − model price)2 RMSE = u . (4.42) t data points data points

We also use Residual standard Error (RSE) for the goodness of ﬁt analysis. This is a statistical measure that is used to describe standard deviation of a point estimate around the ﬁtted function, and this is an estimate of the accuracy of the dependent variable being measured. Math-

39 q SSE ematically, RSE = n−k where n is the number of observations, k is the number of parameters to be estimated, and SSE is the sum of square error.

For the calibration we consider the BN-S model with ρ = −1 in (3.2), so that the Γ and inverse Gaussian models have the same number of parameters as in the Heston model. For the analysis, σ0 is taken to be 0.01. The calibration results for various cases, with the application of Theorem 4.1.2 for variance swap, are shown in Table 1,2, and 3. The corresponding ﬁttings are shown in Figure 1.

The t-value and the probability of each table explains the rejection region under any α level for the signiﬁcance of the variable. In the tables, “P r(> |t|) < 2e − 16” means that the probability of that parameter being zero is less than 2e−16. The column marked “Standard Error” displays the estimated standard errors of these parameter estimate.

Table 4 gives goodness-of-fit comparison for different models. It is clear that the BN-S model with ρ = −1 is producing significant improvement in the root mean square error, than the Heston model with the same parameter. From table (4.3), we can see that the PTS has four parameters, which is one more parameter than Inverse Gaussian, Variance Gamma or Heston model. The improvement of the error is even significant shown in table (4.4).

Table 4.1. Parameter estimate of Gamma distribution for variance swap

Parameters Estimate Standard Error t Value P r(> |t|) ν 0.0065283 0.0005311 12.29 < 2e − 16 λ 0.1092836 0.0052738 20.72 < 2e − 16 α 0.1603521 0.0087384 -18.35 < 2e − 16

Table 4.2. Parameter estimate of Inverse Gaussian for variance swap

Parameters Parameter estimate Standard Error t Value P r(> |t|) λ 0.109283 0.005274 20.72 < 2e − 16 γ 0.023807 0.0010678 -54.90 < 2e − 16 δ 0.586191 0.01039 22.90 < 2e − 16

We use analysis of variance to establish the best model used to estimate the variance swap.

40 Table 4.3. Parameter estimate of Positive tempered stable for variance swap

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.109283 0.005274 20.7 < 2e − 16 δ 0.023807 0.001039 22.86 < 2e − 16 κ .456078 0.005483 24.98 < 2e − 16 γ 0.586191 0.010678 54.90 < 2e − 16

For this, R2 is calculated for each model. The quantity R2 is used to estimate the percentage of the given data that can be explained by the model. It is found that the IG and Γ model explain

81.05% and 82.65% of the data respectively, and all parameters in these cases turn out to be highly signiﬁcant. On the other hand, the PTS model uses one more parameter. The R2 value of the

PTS is found to be around 85.05%. Heston model and Hull-White model explain about 72.85% and 65.28% of the given data respectively.

Table 4.4. Comparing errors of diﬀerent models for variance swap

Model RMSE RSE APE AAE ARPE Hull-White 0.9610284 0.518945 0.00854231 0.0296473 1.9821059 Heston Model 0.0588036 0.0791205 0.000785167 0.59817601 1.0516412 Variance Gamma 0.00229305 0.00105 0.000005095391 0.002303117 0.1287677 Inverse Gaussian 0.00221305 0.00163 0.000005095203 0.002303032 0.1287595 Positive tempered stable 0.001002722 0.0001006 0.0000005106 0.0001350163 0.013285

Again, for the calibration of the volatility swap, we consider the BN-S model with ρ = −1 in (3.2), so that the Γ and inverse Gaussian models have same number of parameters as in the

Heston model. The calibration results for various cases with the application of Theorem 4.2.2 for approximate volatility swap, are shown in Table 5, 6, and 7. Their corresponding fittings are shown in Figure 2. Table 8 gives a comparison of goodness-of-fit for different models. It is clear that the

BN-S models with ρ = −1 are producing better result than the Heston and Hull-White models, which both models are driven by a Brownian motion.

Next, we compare the results for the volatility swap for diﬀerent models and estimate the parameters along with their standard errors. Using statistical analysis at the α level of 0.00001, it turns out that all the model parameters are highly signiﬁcant to predict the true value of the

41 Heston Model Variance Swap Gamma Variance Swap

delivery Price delivery price 0.05 Model Fit 0.05 Model Fit 0.04 0.04 0.03 0.03 Delivery Price Delivery Price 0.02 0.02

0 200 400 600 800 0 200 400 600 800

Maturity(Days) Maturity(Days)

Inverse Gaussian Variance Swap Positive Tempered Stable Variance Swap

delivery price delivery price 0.05 Model Fit 0.05 Model Fit 0.04 0.04 0.03 0.03 Delivery Price Delivery Price 0.02 0.02

0 200 400 600 800 0 200 400 600 800

Maturity(Days) Maturity(Days)

Figure 4.1. Fitting of variance swap realized volatility. Analysis with R2 reveals that more than 83.79% of the data can be explained by Γ model, and more than 83.60% of the data can be explained by IG model. For both cases all the parameters are highly signiﬁcant. On the other hand, the PTS model uses one additional parameter and as a consequence gives a better goodness-of-ﬁt estimate. From the R2 value it is found that around 87.54% of the given data is explained by PTS model. The Heston model and the Hull-White model explain about 68.83% and 63.45% of the given data respectively.

Finally, we consider he calibration for the BN-S model with ρ = −1 using Theorem 4.3.3.

The calibration parameters for fair delivery price of volatility swaps are shown in Table 9, 10, and

11. The corresponding ﬁttings are shown in Figure 3. Table 12 gives a comparison of goodness-of-ﬁt

42 Table 4.5. Parameter estimates for Γ volatility swap with Theorem 4.2.2

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.1845832 0.0085634 21.55 < 2e − 16 ν 0.0093435 0.0002426 38.511 < 2e − 16 α 0.8902156 0.0202934 43.87 < 2e − 16

Table 4.6. Parameter estimates for IG volatility swap with Theorem 4.2.2

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.483033 0.027331 17.649 < 2e − 16 δ 0.038029 0.001266 30.052 < 2e − 16 γ 1.577838 0.027993 56.38 < 2e − 16 for diﬀerent models.

It is clear that the expression in Theorem 4.3.3 is a formal asymptotic expansion. However,

Theorem 4.3.4 and Theorem 4.3.5 gives control over the convergence of the inﬁnite series. Theorem

4.3.6 can be used in practice to demonstrate the reliability of the approximation. It is clear from the above results and the proof of Theorem 4.3.6, that in order to have rapid convergence of the inﬁnite

(n1) (n2) series for E(σR), the quantity max0

(N+1) n1 = N + 1, to demonstrate the rapid convergence of the expression max0

4.5. Conclusion

In this Chapter we have presented a new approach based on the BN-S model to obtain the arbitrage-free pricing for variance and volatility swaps for ﬁnancial markets. The stochastic volatility models used for analysis are empirically reasonable,and the many appealing features from a ﬁnance perspective. The results derived in this dissertation are potentially important as this means that stochastic volatility models built out of OU processes with gamma or inverse

Gaussian or positive tempered stable marginals have excellent numerical accuracy in obtaining the fair delivery prices for various swaps. Further, we get closed form pricing formulas depending on various cumulants of the BDLP L´evyprocess Z. In this dissertation we have also derived an

43 Table 4.7. Parameter estimates for PTS volatility swap with Theorem 4.2.2

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.483033 0.027406 17.64 < 2e − 16 δ 0.038029 0.001266 30.05 < 2e − 16 κ 0.61204 0.0054186 40.05 < 2e − 16 γ 1.577838 0.0027993 56.38 < 2e − 16

Table 4.8. Comparing errors of diﬀerent models for volatility swap

Model RMSE RSE APE AAE ARPE Hull-White 0.8245997 0.721024 0.0001446324 0.076537383 0.5889565 Heston Model 0.245611 0.1746 0.0005610508 0.253595 1.887359 Variance Gamma 0.007656574 0.005441 0.00001886869 0.008528647 0.06435461 Inverse Gaussian 0.0077152225 0.005483 0.00001867513 0.008441158 0.06352462 Positive tempered stable 0.00021903 0.000163 0.00001867513 0.00844115 0.006352462 algorithmic process to compute the cumulants of Z. The improvement of numerical results in the analysis is very signiﬁcant over the existing models with a similar number of parameters, such as the Heston model.

More generally, we also used numerical approximations and statistical analysis for model adequecy and it turns out that Ou Prcess captured most of the variance and volatiltiy jumps as suppose to Heston and Hull-white model. Those models can be also extended for commodity swap of natural gass and crude oil. The approach towards this model can be considered in future works.

44 Heston Model Volatility Swap Gamma Volatility Swap

delivery price delivery price

0.22 Model Price 0.22 Model Price 0.20 0.20 0.18 0.18 Delivery Price Delivery Price 0.16 0.16 0.14 0.14 0.12 0.12

0 200 400 600 800 0 200 400 600 800

Maturity(Days) Maturity(Days)

Inverse Gaussian Volatility Swap Positive Tempered Stable Volatility Swap

Delivery price delivery price

0.22 Model Price 0.22 Model Price 0.20 0.20 0.18 0.18 Delivery Price Delivery Price 0.16 0.16 0.14 0.14 0.12 0.12

0 200 400 600 800 0 200 400 600 800

Maturity(Days) Maturity(Days)

Figure 4.2. Fitting of volatility swap

Table 4.9. Parameter estimates for Γ volatility swap with Theorem 4.3.3

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.204673 0.003423 59.80 < 2e − 16 ν 0.443597 0.013299 33.35 < 2e − 16 α 1.386975 0.041721 33.24 < 2e − 16

Table 4.10. Parameter estimates for IG volatility swap with Theorem 4.3.3

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.199502 0.003372 59.16 < 2e − 16 δ 0.385977 0.005965 64.70 < 2e − 16 γ 1.206734 0.08757 64.33 < 2e − 16

45 Table 4.11. Parameter estimates for PTS volatility swap with Theorem 4.3.3

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.199554 0.003373 59.17 < 2e − 16 δ 0.385881 0.005962 64.72 < 2e − 16 κ 0.614802 0.001289 64.78 < 2e − 16 γ 1.206434 0.018746 64.36 < 2e − 16

Table 4.12. Comparing errors of diﬀerent models for volatility swap

Model RMSE RSE APE AAE ARPE Variance Gamma 0.007674167 0.005123 0.00001907175 0.009620432 0.0513133 Inverse Gaussian 0.007663909 0.005446 0.00001905593 0.008613282 0.06508503 Positive tempered stable 0.000137236 0.00054102 0.00000127545 0.008620456 0.06513158

Positive Tempered Stable Volatility Swap Gamma Volatility Swap Inverse Gaussian Volatility Swap

delivery price delivery price delivery price Model Price Model Fit Model Fit 0.20 0.20 0.20 0.16 0.16 0.16 Delivery Price Delivery Price Delivery Price 0.12 0.12 0.12

0 200 400 600 800 0 200 400 600 800 0 200 400 600 800

Maturity(Days) Maturity(Days) Maturity(Days)

Figure 4.3. Fitting of volatility swap with improved model

Maximum Error Estimate for Volatility swap Maximum Error Estimate for Volatility swap

● ● IG IG PTS 0.12 PTS VG VG 6e−04 0.10 0.08 4e−04 Error Error 0.06 0.04 2e−04

● 0.02

● ●

● ● ● 0.00 0e+00 1 2 3 4 5 3.0 3.5 4.0 4.5 5.0

N N

Figure 4.4. (Left) Reliability of ﬁtting with truncated inﬁnite series for volatility swap. (Right) Zoomed in version of the left picture.

46 5. PRICING COVARIANCE AND CORRELATION SWAP

FOR TWO ASSETS WITH A STOCHASTIC VOLATILITY

In this chapter we price covariance swaps for ﬁnancial market when it is governed by the BN-

S model. Covariance swaps are recent financial products that are useful for volatility hedging and speculation using two different financial underlying assets. Introduced in [20] this contract pays the excess of the realized covariance between two derivatives over a constant specified at the outset of the contract. Such a contract may serve as a useful complement for the variance contracts that trade over the counter on several currencies. This makes covariance swap an over the counter financial derivative that allows one to speculate on or hedge risks with the magnitude of the movement, i.e. volatility of the underlying assets like exchange rate, interest rate, or stock index. For example, options dependent on exchange rate movements have an exposure to movements of the correlation between the asset and the exchange rate. This risk may be hedged by using covariance swap. The analysis of a contract paying the realized covariance is a necessary precursor for the analysis of further derivatives written on covariance.

5.1. Pricing Covariance Swap

1 Deﬁnition 5.1.1. The covariance swap is a covariance forward contract between two assets (St 2 and St , 0 ≤ t ≤ T ) of a realized covariance, and its payoﬀ at maturity is given by

1 2 N(CovR(S ,S ) − KCov), (5.1)

1 2 where N is a notional amount, CovR(S ,S ) is the realized covariance of two given assets and KCov is the strike price.

The arbitrage free price of the covariance swap (over the period 0 ≤ t ≤ T ) is given by

−rT 1 2 PCov = E e (CovR(S ,S ) − KCov) , (5.2) where E(·) is the expectation with respect to some equivalent martingale measure. In this section we implement a generalized version of the Barndorﬀ-Nielsen and Shephard model to model one

47 ∗ of the two assets for covariance and correlation swaps. Let Zt and Zt be two independent Lévy ∗ subordinators. Here the independence of the Lévyprocesses Zt and Zt is understood in the sense of [63] (Proposition 5.3). That is, if (Xt,Yt) is a Lévy process with Lévymeasure ν(X,Y ) and without Gaussian part, then its components are independent if and only if the support of ν(X,Y ) is contained in the set {(x, y): xy = 0}, that is, if and only if they never jump together. In this case

ν(X,Y )(A) = νX (AX ) + νY (AY ), where AX = {x :(x, 0) ∈ A} and AY = {y : (0, y) ∈ A}, and νX and νY are Lévymeasures of Xt and Yt. In this case, we define a subordinator which a linear combination of the above to Lévy processes as

˜ 0 p 02 ∗ dZt = ρ dZt + 1 − ρ dZt , (5.3)

0 0 provided 0 ≤ ρ ≤ 1. Thus, for 0 ≤ ρ ≤ 1, Zt and Z˜t are positively correlated L´evysubordinators

(see [55]). It is clear that Var(Zt) = tκ2, Var(Z˜t) = tκ˜2, and using the linearity of covariance and

˜ 0 p 02 ∗ Zt = ρ Zt + 1 − ρ Zt gives

˜ 0 p 02 ∗ Cov(Zt, Zt) = Cov(Zt, ρ Zt + 1 − ρ Zt )

0 p 02 ∗ = (Cov)(Zt, ρ Zt) + Cov(Zt, 1 − ρ Zt )

0 = ρ (Cov)(Zt,Zt)

0 = ρ tκ2. (5.4)

The last step is from independence and κ2 andκ ˜2 are the variances (second cumulant) of

Z1 and Z˜1 respectively. Therefore the correlation coeﬃcient between Zt and Z˜t is independent of time t and is given by

Cov(Zt, Z˜t) Corr(Zt, Z˜t) = p q Var(Zt) Var(Z˜t) ρ0tκ = √ √2 tκ2 tκ˜2 rκ = ρ0 2 . (5.5) κ˜2

i Xi Assume that the risk-neutral dynamics of the two assets are given by St = e t , with i = 1, 2,

48 i where the stochastic processes Xt are driven by a linear combination of a Weiner process and a L´evyprocess as given by:

i i i i (i) dXt = bt dt + (σ )t dWt + ρ dZλt, i = 1, 2, (5.6) and

1 2 1 2 1 2 d(σ )t = −λ(σ )t dt + dZλt, (σ )0 > 0, (5.7) and 2 2 2 2 ˜ 2 2 d(σ )t = −λ(σ )t dt + dZλt, (σ )0 > 0, (5.8)

1 2 1 2 1 2 where Wt and Wt are correlated Wiener processes with Cov(Wt ,Wt ) =ρt ¯ , and bt and bt are 1 2 2 2 (1) (2) deterministic functions of (σ )t and (σ )t respectively. In (5.6) ρ and ρ are leverage parameters corresponding to S1 and S2 respectively.

1 2 From the deﬁnition of CovR(S ,S ) we have

1 1 Cov (S1,S2) = [ln S1 , ln S2 ] = [X1 ,X2 ], (5.9) R T T T T T T where [·, ·] represents the quadratic covariation.

We make the following assumption for the L´evysubordinator Zt.

Assumption 4. Zt is a L´evysubordinator with ﬁnite variation and no deterministic drift. There- fore, if JZ is the random measure describing jumps of Z then

Z t Z ∞ Zt = yJZ (ds, dy). (5.10) s=0 0

∗ Remark 5.1.2. If the L´evymeasures of Z and Z are νZ and νZ∗ respectively, then by Assumption ˜ ˜ ˜ 1 and [63] (Theorem 4.1), the characteristic triplet of Z is given by (A, γ,˜ νZ˜), where A = 0, B B ν ˜(B) = νZ 0 + νZ∗ √ , for B ∈ B( ) and Z ρ 1−ρ02 R

Z 0 p 02 ∗ 0 p 02 ∗ γ˜ = ρ γ + 1 − ρ γ − y 1|y|≤1(y) − 1S1 (y) νZ˜(dy) = ρ γ + 1 − ρ γ , (5.11) R

0 p 02 2 2 where S1 is given by S1 = {ρ x1 + 1 − ρ x2|x1 + x2 ≤ 1, x1, x2 ∈ R}. Therefore in general

49 Z˜ has a drift component. However, if both Z and Z∗ satisfy Assumption 4, i.e., if both Z and

∗ R ∗ R Z are processes of ﬁnite variation and γ = |x|≤1 xνZ (dx) and γ = |x|≤1 xνZ∗ (dx), then γ˜ = R ∗ |x|≤1 xνZ˜(dx) and hence the deterministic drift (in the sense of Corollary 3.1 in [63]) for Z, Z ˜ ˜ and Z is zero. Therefore, if JZ˜ is the random measure describing jumps of Z then

Z t Z ∞ ˜ Zt = yJZ˜(ds, dy). (5.12) s=0 0

From (5.6) we obtain

Z T Z T Z λT Z ∞ i i i i (i) XT = bt dt + (σ )t dWt + ρ yJZ (ds, dy), i = 1, 2. (5.13) 0 0 0 0

1 2 Therefore the quadratic covariation of XT and XT is given by (see [63], Section 8.2.2)

Z T Z λT Z ∞ 1 2 1 2 (1) (2) 2 [XT ,XT ] = ρ¯(σ )t(σ )t dt + ρ ρ y JZ (ds, dy). (5.14) 0 0 0

It is clear from (5.2) and (5.9) that to ﬁnd the arbitrage free price of covariance swap it

1 2 1 1 2 is sufficient to compute E(CovR(S ,S )) = T E[XT ,XT ]. We proceed to find this in the following results. The first lemma is similar to Lemma 4.3.2

R λt s Lemma 5.1.3. Suppose that Bt = α1 + 0 e dVs, where λ > 0, α1, λ ∈ R are constants, and 0 ≤ t ≤ T , and V is a L´evysubordinator with no deterministic drift. Then

Z λt s ΦBt (θ) = exp iθα1 + κV1 (iθe ) ds , (5.15) 0

where κV1 (·) is the cumulant generating function for V1. The moments of Bt are given by

k k E(Bt ) = (−i) g˜k(0), k = 1, 2,..., (5.16) where Z λt s 0 s g˜1(θ) = i α + e κV1 (iθe ) ds , (5.17) 0

50 and

0 g˜k+1(θ) =g ˜1(θ)˜gk(θ) +g ˜k(θ), k = 1, 2,.... (5.18)

In the above formulas prime represents the derivative with respect to the parameter in parenthesis.

Proof. The proof is similar to that of Lemma 4.3.2 and we omit the details here.

˜ R λt s R λt s ˜ When Vt = Zt, or Vt = Zt, any moment for Bt = α1 + 0 e dZs (or, Bt = α1 + 0 e dZs) can be obtained in terms of κm, m = 1, 2,... , which are cumulants of Z1 (or, in terms ofκ ˜m, ˜ R λt s m = 1, 2,... , which are cumulants of Z1). For example with Bt = α1 + 0 e dZs,

Z λt s λt E(Bt) = −ig˜1(0) = α1 + κ1e ds = α1 + κ1(e − 1), (5.19) 0 and

2 2 2 0 E(Bt ) = (−i) g˜2(0) = −(˜g1(0) +g ˜1(0)) Z λt 2 Z λt s 2s = α + κ1e ds + κ2e ds 0 0 2 κ = α + κ (eλt − 1) + 2 (e2λt − 1). (5.20) 1 2

R λt s ˜ Similar results hold for Bt = α1 + 0 e dZs, with κ1 and κ2 replaced byκ ˜1 andκ ˜2 respectively. 1 2 Before proceeding further we prove a result related to the correlation coeﬃcient of (σ )t 2 2 ˜ ∗ and (σ )t . We show that if Z, Z, and Z are related by (5.3) then the correlation coeﬃcient of 1 2 2 2 (σ )t and (σ )t is independent of t.

Theorem 5.1.4. Suppose Z, Z˜, and Z∗ are related by (5.3). Then the correlation coeﬃcient of q (σ1)2 and (σ2)2 is independent of time and is given by ρ0 κ2 , where κ and κ˜ are second cumulant t t κ˜2 2 2 of Z1 and Z˜1 respectively.

Proof. It is clear from (5.7) and (5.8) that

Z λt 1 2 −λt 1 2 s (σ )t = e (σ )0 + e dZs , (5.21) 0

51 and Z λt 2 2 −λt 2 2 s ˜ (σ )t = e (σ )0 + e dZs . (5.22) 0

Therefore by Lemma 5.1.3, we can obtain

Z λt 1 2 −2λt s −2λt κ2 2λt Var((σ )t ) = e Var e dZs = e (e − 1), (5.23) 0 2 and Z λt 2 2 −2λt s ˜ −2λt κ˜2 2λt Var((σ )t ) = e Var e dZs = e (e − 1). (5.24) 0 2

Note that

Z λt 2 2 0 1 2 −λt 2 2 0 1 2 p 02 s ∗ (σ )t = ρ (σ )t + e ((σ )0 − ρ (σ )0) + 1 − ρ e dZs . (5.25) 0

Therefore using the independence of Z and Z∗, we obtain

ρ0κ Cov((σ1)2, (σ2)2) = ρ0Var((σ1)2) = e−2λt 2 (e2λt − 1). (5.26) t t t 2

Hence the theorem follows from (5.23), (5.24), and (5.26).

For proceeding further with the arbitrage free pricing for covariance swap, we assume that

1 2 2 2 2 2 for 0 ≤ t ≤ T ,(σ )t < β1 and (σ )t < β2 , for some β1, β2 > 0. We take β = max{β1, β2}. This is a 1 2 very reasonable assumption. For example, since (σ )t and (σ )t are expressed in percentages, for a normal market, where it does not have “crash-like ﬂuctuations” β = 1 can be assumed. With this deﬁnition of β we state the next lemma.

1 2 2 2 ˜ Lemma 5.1.5. Suppose that (σ )t and (σ )t are given by (5.7) and (5.8) respectively, and Z, Z ∗ 1 2 2 2 2 and Z are related by (5.3). Let there exists β > 0 such that for 0 ≤ t ≤ T , (σ )t , (σ )t < β . Then

1 2 E (σ )t(σ )t

∞ k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) = e (β) ρ (1 − ρ ) 2 N (p, u), (5.27) 4k(k!)2(2k − 1) p u t k=0 p=0 u=0

52 where

p−u Z λt p+u 2 2 0 1 2 Z λt ! 1 2 s (σ )0 − ρ (σ )0 s ∗ Nt(p, u) = E (σ )0 + e dZs E p + e dZs , (5.28) 0 1 − ρ02 0 can be computed using Lemma 5.1.3.

Proof. It is clear from (5.7) and (5.8) that

Z t 1 2 R λt s ! 1 2 −λt 1 2 −λt λs −λt 2 (σ )0 + 0 e dZs (σ )t = e (σ )0 + e e dZλs = e β 2 , (5.29) 0 β and

Z t 2 2 −λt 2 2 −λt λs ˜ (σ )t = e (σ )0 + e e dZλs 0 2 2 0 R λt s p 02 R λt s ∗ ! (σ ) + ρ e dZs + 1 − ρ e dZ = e−λtβ2 0 0 0 s . (5.30) β2

1 2 R λt s 0 2 2 0 1 2 R λt s ∗ We denote Ft = (σ )0 + 0 e dZs, c = (σ )0 −ρ (σ )0 and Gt = 0 e dZs and Therefore we obtain

s ρ0F 2 + c0F + p1 − ρ02F G (σ1) (σ2) = e−λtβ2 t t t t t t β4 s ρ0F 2 + c0F + p1 − ρ02F G − β4 = e−λtβ2 1 + t t t t . (5.31) β4

By the construction we have

ρ0F 2 + c0F + p1 − ρ02F G − β4 t t t t < 1. (5.32) 4 β

Using (4.18) we obtain

53 ∞ X (−1)k+1(2k)! 1 p k (σ1) (σ2) = e−λt ρ0F 2 + c0F + 1 − ρ02F G − β4 t t 4k(k!)2(2k − 1) β4k−2 t t t t k=0 ∞ k k+1 p X (−1) (2k)! X k p p = e−λt F ρ0F + c0 + 1 − ρ02G (−β)4k−4p 4k(k!)2(2k − 1)β4k−2 p t t t k=0 p=0 ∞ k+1 k p X (−1) (2k)! X k p X p p = e−λt F ρ0uF u(c0 + 1 − ρ02G )p−u(−β)4k−4p 4k(k!)2(2k − 1) p t u t t k=0 p=0 u=0 ∞ k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) p+u p−u = e (β) ρ (1 − ρ ) 2 F G˜ (5.33) 4k(k!)2(2k − 1) p u t t k=0 p=0 u=0

c0 ∗ where G˜t = Gt + √ . Since Z and Z are independent, we note that Ft and G˜t are independent. 1−ρ02 Therefore

1 2 E (σ )t(σ )t =

∞ k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) p+u p−u = e (β) ρ (1 − ρ ) 2 E F E G˜ . (5.34) 4k(k!)2(2k − 1) p u t t k=0 p=0 u=0

p+u 1 2 We note that E(Ft ) in (5.34) can be computed by Lemma 5.1.3 where α1 = (σ )0 and Vt = Zt. 2 2 0 1 2 p−u (σ )0−ρ (σ )0 Similarly E(G˜ ) in (5.34) can be computed by Lemma 5.1.3 where α1 = √ and Vt = t 1−ρ02 ∗ Zt .

Next we consider the inﬁnite series

∞ k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) e (β) ρ (1 − ρ ) 2 N (p, u) (5.35) 4k(k!)2(2k − 1) p u t k=0 p=0 u=0

√ 0 2 0 02 4 of Lemma 5.1.5. It is clear from the proof of that Lemma that ρ Ft +c Ft+ 1−ρ FtGt−β < 0. It is √ β4 0 2 0 02 4 ρ Ft +c Ft+ 1−ρ FtGt−β also clear that 4 < 1. β

1 2 Lemma 5.1.6. The quantity E((σ )t(σ )t) can be approximated by the n-th partial sum

n−1 k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) e (β) ρ (1 − ρ ) 2 N (p, u), (5.36) 4k(k!)2(2k − 1) p u t k=0 p=0 u=0

54 with the absolute error of approximation less than the quantity e−λtβ2 1√ , for n ≥ 1. (2n−1) 3n+1

Proof. Observe that

ρ0F 2 + c0F + p1 − ρ02F G − β4 t t t t < 1, (5.37) 4 β and

∞ X (−1)k+1(2k)! 1 p k (σ1) (σ2) = e−λtβ2 ρ0F 2 + c0F + 1 − ρ02F G − β4 . (5.38) t t 4k(k!)2(2k − 1) β4k t t t t k=0

The rest of the proof is similar to the proof of Theorem 4.3.4.

Remark 5.1.7. The quantity β can be used as a “control parameter” that improves the rate of convergence of the inﬁnite series

∞ k p p+1 −λt X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) e (β) ρ (1 − ρ ) 2 N (p, u). (5.39) 4k(k!)2(2k − 1) p u t k=0 p=0 u=0

√ 0 2 0 02 4 ρ Ft +c Ft+ 1−ρ FtGt−β 1 Suppose that it is possible to choose β so that 4 < , for some > 0 and β 2+ 0 ≤ t ≤ T . Then a procedure analogous to the proof of Theorem 4.3.5 can be used to show that

1 2 the quantity E((σ )t(σ )t) can be approximated by the n-th partial sum of the inﬁnite series (5.39), with the absolute error of approximation less than the quantity

1 1 + 2 1 1 e−λtβ2 √ , for n ≥ 1. (5.40) 2 + (2n − 1) 3n + 1 (1 + )n

Also, we note that in this context it is possible to derive a result analogous to Theorem 4.3.6.

Now we prove the main theorem for this section. As in Remark 5.1.2 we denote the L´evy measure for Z by νZ .

Theorem 5.1.8. The arbitrage-free value of a covariance swap is given by

−rT PCov = e (g1(T ) + g2 − KCov) , (5.41)

55 where

∞ k p p+1 ρ¯ X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) g (T ) = (β) ρ (1 − ρ ) 2 L(p, u), (5.42) 1 T 4k(k!)2(2k − 1) p u k=0 p=0 u=0

R T −λt where L(p, u) = 0 e Nt(p, u) dt with Nt(p, u) given by (5.28), and

(1) (2) g2 = ρ ρ λκ2. (5.43)

Proof. It is suﬃcient to show that under the equivalent martingale measure

1 2 E(Cov(ST ,ST ) = g1(λ, T ) + g2(λ, T ). (5.44)

From (5.14) we obtain

1 E(Cov (S1,S2)) = E[X1 ,X2 ] R T T T Z T (1) (2) Z λT Z ∞ 1 1 2 ρ ρ 2 = ρE¯ (σ )t(σ )t dt + E y JZ (ds, dy) . (5.45) T 0 T 0 0

Using Lemma 5.1.5 we obtain

Z T 1 1 2 ρE¯ (σ )t(σ )t dt T 0 ∞ k p p+1 Z T ρ¯ X X X (−1) (2k)! k p −4p+2 0u 02 (p−u) −λt = (β) ρ (1 − ρ ) 2 e N (p, u) dt, (5.46) T 4k(k!)2(2k − 1) p u t k=0 p=0 u=0 0 where Nt(p, u) is given by (5.28).

Observing E(JZ (ds, dy)) = νZ (dy) ds, we obtain

(1) (2) Z λT Z Z ∞ ρ ρ 2 (1) (2) 2 E y JZ (ds, dy) = ρ ρ λ y νZ (dy). (5.47) T 0 R 0

However since Zt is a subordinator, therefore (see [63], Proposition 3.13) we obtain that

R ∞ 2 κ2 = Var(Z1) = 0 y νZ (dy). Hence the theorem is proved.

56 5.2. Pricing Correlation Swap

Deﬁnition 5.2.1.

We can immediately derive a corollary of Theorem 5.1.8 related to the correlation swap.A correlation swap is a forward contract on the correlation between the underlying assets S1 and S2 for which payoﬀ at maturity is equal to

1 2 N(CorrR(S ,S ) − KCorr), (5.48)

1 2 where KCorr is the strike price, N is the notional amount and CorrR(S ,S ) is the realized correlation deﬁned by 1 2 1 2 CovR(S ,S ) CorrR(S ,S ) = q q . (5.49) 1 2 2 2 (σ )R (σ )R

The arbitrage free value of the correlation swap is given by

−rT 1 2 PCorr = e E CorrR(S ,S ) − KCorr . (5.50)

1 2 2 2 1 2 Corollary 5.2.2. Suppose that (σ )R and (σ )R are realized variances of S and S respectively over the time interval [0,T ]. Then the arbitrage-free value of a correlation swap can be approximated by   −rT g1(T ) + g2 PCorr ≈ e q q − KCorr , (5.51) 1 2 2 2 E(σ )R E(σ )R where g1(T ) and g2 can be obtained from Theorem 5.1.8, and

1 2 E(σ1)2 = λ−1(1 − e−λT )(σ1)2 + κ (T − λ−1(1 − e−λT )) + ρ(1) λκ , (5.52) R T 0 1 2

1 2 E(σ2)2 = λ−1(1 − e−λT )(σ2)2 +κ ˜ (T − λ−1(1 − e−λT )) + ρ(2) λκ . (5.53) R T 0 1 2

1 2 Proof. As both Xt and Xt are driven by Z, the realized volatility are given by

Z T 2 1 2 1 1 2 (1) (σ )R = (σ )t dt + ρ λκ2, (5.54) T 0

57 and Z T 2 2 2 1 2 2 (2) (σ )R = (σ )t dt + ρ λκ2. (5.55) T 0

1 2 2 2 Thus the expressions of E(σ )R and E(σ )R can be obtained from (4.3). It follows from [50] that

1 2 1 2 E CovR(S ,S ) E(CorrR(S ,S )) ≈ q q . (5.56) 1 2 2 2 E(σ )R E(σ )R

Hence the corollary follows from Theorem 5.1.8.

We conclude the theoretical part of this section by an alternative approximate version of

Theorem 5.1.8. For that we need the following lemmas.

Lemma 5.2.3. Suppose Z, Z˜, and Z∗ are related by (5.3). For the volatility dynamics (5.7) of

1 2 two assets St and St ,

1 2 2 2 E (σ )t (σ )t

−2λt 1 2 2 2 1 2p 02 ∗ 0 1 2 2 2 −2λt λt = e (σ )0(σ )0 + (σ )0 1 − ρ κ1 + (ρ (σ )0 + (σ )0)κ1 e (e − 1)

κ2 p + e−2λt ρ0 κ2(eλt − 1)2 + (e2λt − 1) + 1 − ρ02κ κ∗(eλt − 1)2 , (5.57) 1 2 1 1

∗ ∗ and where κ1 and κ2 are ﬁrst two cumulants of Z1, and κ1 is the ﬁrst cumulant of Z1 .

Proof. It is clear from (5.7) and (5.8) that

Z t Z t 1 2 2 2 −2λt 1 2 2 2 −2λt 1 2 λs ˜ 2 2 λs (σ )t (σ )t = e (σ )0(σ )0 + e (σ )0 e dZλs + (σ )0 e dZλs 0 0 Z t Z t −2λt λs λs + e e dZλs e dZ˜λs. (5.58) 0 0

˜ ∗ Note that the ﬁrst cumulant of Z1 (denoted asκ ˜1) is related to those of Z1 and Z1 by the 0 p 02 ∗ simple relationκ ˜1 = ρ κ1 + 1 − ρ κ1. Therefore we observe

Z t λs λt E e dZλs = κ1(e − 1), (5.59) 0

58 and

Z t λs ˜ 0 p 02 ∗ λt E e dZλs = (ρ κ1 + 1 − ρ κ1)(e − 1). (5.60) 0

Using (5.3) and the independence of L´evyprocesses Z and Z∗, we obtain

Z t Z t λs λs E e dZλs e dZ˜λs 0 0 Z t 2 Z t Z t 0 λs p 02 λs λs ∗ = ρ E e dZλs + 1 − ρ E e dZλs E e dZλs 0 0 0 κ2 p = ρ0 κ2(eλt − 1)2 + (e2λt − 1) + 1 − ρ02κ κ∗(eλt − 1)2. (5.61) 1 2 1 1

Hence (5.57) follows from (5.58) and the above results.

To keep track of constants for the next computations, we deﬁne the followings:

1 2 2 2 a1 = (σ )0(σ )0, (5.62)

1 2p 02 a2 = (σ )0 1 − ρ , (5.63)

2 2 0 1 2 a3 = (σ )0 + ρ (σ )0, (5.64)

0 a4 = ρ , (5.65) and

p 02 a5 = 1 − ρ . (5.66)

Using Lemma 5.1.3 we can compute the following:

Z t 3 λs 3 λt 3 3κ1κ2 λt 2λt κ3 3λt E e dZλs = κ1(e − 1) + (e − 1)(e − 1) + (e − 1), (5.67) 0 2 3

59 and

Z t 4 λs 4 λt 4 2 λt 2 2λt E e dZλs = κ1(e − 1) + 3κ1κ2(e − 1) (e − 1) 0 3 4 κ + κ2(e2λt − 1)2 + κ κ (eλt − 1)(e3λt − 1) + 4 (e4λt − 1). (5.68) 4 2 3 1 3 4

R t λs R t λs ∗ Let us deﬁne X1t = 0 e dZλs and X2t = 0 e dZλs. For notational convenience we will simply write X1t = X1 and X2t = X2. Clearly X1 and X2 are independent. Thus from (5.58) it can be easily shown that

1 2 2 2 2 Var (σ )t (σ )t = Var(a1 + a2X2 + a3X1 + a4X1 + a5X1X2)

2 2 2 2 2 = a2Var(X2) + a3Var(X1) + a4Var(X1 ) + a5Var(X1X2)

2 + 2a2a5Cov(X2,X1X2) + 2a3a4Cov(X1,X1 )

2 + 2a3a5Cov(X1,X1X2) + 2a4a5Cov(X1 ,X1X2). (5.69)

Using the independence of X1 and X2, it is easy to show

2 4 2 2 Var(X1 ) = E(X1 ) − (E(X1 )) , (5.70)

2 2 2 2 Var(X1X2) = E(X1 )E(X2 ) − (E(X1)) (E(X2)) , (5.71)

Cov(X2,X1X2) = E(X1)Var(X2), (5.72)

2 3 2 Cov(X1,X1 ) = E(X1 ) − E(X1 )E(X1), (5.73)

Cov(X1,X1X2) = E(X2)Var(X1), (5.74)

2 3 2 Cov(X1 ,X1X2) = E(X2)(E(X1 ) − E(X1 )E(X1)). (5.75)

60 Clearly,

λt ∗ λt E(X1) = κ1(e − 1),E(X2) = κ1(e − 1), (5.76)

κ κ∗ E(X2) = κ2(eλt − 1)2 + 2 (e2λt − 1),E(X2) = (κ∗)2(eλt − 1)2 + 2 (e2λt − 1), (5.77) 1 1 2 2 1 2 and κ κ∗ Var(X ) = 2 (e2λt − 1), Var(X ) = 2 (e2λt − 1). (5.78) 1 2 2 2

3 4 The quantities E(X1 ) and E(X1 ) can be computed using (5.67) and (5.68) respectively. 1 2 2 2 Thus (5.69) can be used to construct the variance of (σ )t (σ )t . We conclude the section with following approximation theorem.

Theorem 5.2.4. The arbitrage-free value of a covariance swap is given by

−rT PCov ≈ e (g3(T ) + g2 − KCov) , (5.79)

(1) (2) where g2 = ρ ρ λκ2, and

ρ¯ Z T q Var[(σ1)2(σ2)2] g (T ) ≈ E[(σ1)2(σ2)2] − t t dt, (5.80) 3 t t 1 2 2 2 3/2 T 0 8(E[(σ )t (σ )t ])

1 2 2 2 1 2 2 2 where E[(σ )t (σ )t )] can be computed using Lemma 5.2.3 and Var[(σ )t (σ )t )] can be obtained using (5.69).

Proof. We observe that,

q Var[(σ1)2(σ2)2] E (σ1) (σ2) ≈ E[(σ1)2(σ2)2] − t t . (5.81) t t t t 1 2 2 2 3/2 8(E[(σ )t (σ )t ])

The rest of the proof follows from Theorem 5.1.8.

5.3. Model Fitting and Parameter Estimate

We use the stock prices of S&P500 index and NASDAQ during the time period 10/01/2010 through 01/15/2015. The data set has 1080 closing daily stock prices, and these are used for the computation of realized covariance. For the numerical simulation we take ρ(1) = ρ(2) = −1. From

61 the regression fit model for non linear least square estimate we find that the α level of the parameters for all cases are significantly less than 0.05. Therefore this tells us all the parameters are significantly important to estimate the realized covariance swap price. Once calibration is performed over the described historical data set, we obtain the model parameters and these parameters can be used to price the fixed leg (fair delivery price) of the covariance swap. The calibration results with the application of Theorem 5.1.8 for covariance swap for various cases are shown in Tables 5.1, 5.2 and

5.3 and the corresponding ﬁttings are shown in Figure 5.1. Moreover Figure 5.2 deals with the error and rate of convergence for theorem (5.1.8) and it clearly shows how fast the convergence existst as n → ∞.

1 2 For Table 5.1 it is assumed that for the unconditional distribution (σ )t ∼ Γ(ν1, α1) and 2 2 1 2 (σ )t ∼ Γ(ν2, α2). For Table 5.2 it is assumed that for the unconditional distribution (σ )t ∼ 2 2 IG(δ1, γ1) and (σ )t ∼ IG(δ2, γ2). Finally for Table 5.3 it is assumed that for the unconditional 1 2 2 2 distribution (σ )t ∼ PTS(κ1, δ1, γ1) and (σ )t ∼ PTS(κ2, δ2, γ2). It is clear that the expression in Theorem 5.1.8 is a formal asymptotic expansion. However,

Remark 5.1.7 gives a control over the convergence of the inﬁnite series. This can also be used in practice to demonstrate the reliability of the approximation. A similar argument as in the case of volatility swap and Remark 5.1.7 show that in order to have rapid convergence of the inﬁnite series in context, the quantity max0

5.4. Conclusion

In this Chapter we have presented a new extended approach based on the BN-S model to obtain the arbitrage-free pricing for covariance and correlation swaps for financial markets. We used the S&P500 index and NASDAQ for numerical purposes and model fitting. Covariance and correlation swap are important in minimizing the risk that might come from different currencies.

We use an OU process driven by a subordinator.

The stochastic volatility models that are used for analysis are empirically reasonable as well as having many appealing features from a theoretical ﬁnance perspective. The results derived in this dissertation are potentially important as this means that stochastic volatility models built

62 out of OU processes with gamma or inverse Gaussian or positive tempered stable marginals have excellent numerical accuracy in obtaining the fair delivery prices for various swaps. Further, we get closed form pricing formulas depending on various cumulants of the BDLP L´evyprocess Z. In this dissertation we have also derived an algorithmic process to compute the cumulants of Z. The improvement of numerical results in the analysis is very signiﬁcant over the existing models with a similar number of parameters, such as the Heston model.

More generally, the results obtained in this dissertation have important implications for their use in, for example, energy markets. Energy is the most important commodity sector. Crude oil and natural gas are one of the most liquid option markets among all commodities. Since Crude oil and natural gas have the properties of mean reverting,which means that they tend to return over time to the long term average.So it is important to estimate or model the direction of price of those types of commodites. Varianceor volatility risk premia for energy commodities, crude oil and natural gas, is becoming increasingly popular and the approach considered in this dissertation can be further developed to analyze such markets. Moreover, the idea used in this dissertation can be generalized for the analysis of covariance and correlation risk of two commodities of an energy market. These aspects will be developed in future works.

63 Table 5.1. Parameter estimates for Variance Gamma Covariance swap

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.15247 0.03418 4.460 9.05e-06 ν1 65.01301 25.08776 2.591 0.009688 ν2 65.01301 25.08776 2.591 0.009688 ρ¯ −0.1415 0.0215 3.219 2e − 8 ρ0 0.038901 0.0014 214.913 < 2e − 16 α1 33.99290 10.25798 3.314 0.000951 α2 19.72202 4.64716 4.244 2.39e − 05

Table 5.2. Parameter estimates for Inverse Gaussian covariance swap

Parameters Estimate Standard Error t Value P r(> |t|) λ 0.15247 0.03418 4.460 9.05e − 06 δ1 11.15092 2.62873 4.242 2.41e − 05 δ2 11.15092 2.62873 4.242 2.41e − 05 ρ¯ −0.38901 0.0215 3.219 2e − 8 ρ0 0.38901 0.0014 214.913 < 2e − 16 γ1 5.83038 0.87971 6.628 5.39e − 11 γ2 3.38266 0.30425 11.118 < 2e − 16

Table 5.3. Parameter estimates for Positive Tempered Stable covariance swap

Parameters Estimate Standard Error t Value P r(> |t|) inserts single horizontal line λ 0.000644975 6.954e − 6 -92.75 < 2e − 16 ρ¯ 91.9824 1.4e − 6 214.913 < 2e − 16 ρ0 2.46078 9.708e − 2 -17.77 < 2e − 16 κ1 0.508837 2.62873 4.242 < 2e − 16 κ2 0.508837 2.53e − 03 1166.18 2e − 16 δ1 3.06699 2.53e − 03 1166.18 2e − 16 δ2 3.06699 2.53e − 03 1166.18 2e − 16 γ1 1.5000 2.148e − 03 793.16 < 2e − 16 γ2 1.98114 0.00030425 212.861 < 2e − 16

Variance Gamma Covariance Swap Inverse Gaussian Covariance Swap Positive Tempered Stable Covariance Swap

0.040 delivery price 0.040 delivery price Model Fit Model Fit delivery price Model Fit 0.035 0.035 0.035 0.030 0.030 0.030 0.025 0.025 0.025 Delivery Price Delivery Price Delivery Price 0.020 0.020 0.020 0.015 0.015 0.015

0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000

Maturity(Days) Maturity(Days) Maturity(Days)

Figure 5.1. Fitting of covariance swap

64 Maximum Error Estimate for Covariance swap Maximum Error Estimate for Covariance swap 0.06

0.7 IG IG PTS PTS ● VG VG 0.6 0.05 0.5 0.04 ● 0.4 0.03 Error Error 0.3 0.02 0.2 0.01 0.1 ● ● ● ● ● 0.0 0.00

2.0 2.5 3.0 3.5 4.0 4.5 5.0 3.0 3.5 4.0 4.5 5.0

N N

Figure 5.2. (Left) Reliability of ﬁtting with truncated inﬁnite series for covariance swap. (Right) Zoomed in version of the left picture.

65 6. ORNSTEIN-UHLENBECK PROCESS FOR

GEOPHYSICAL DATA ANALYSIS

6.1. Introduction

Earthquake occurs in a region where there is a multiscale networks or systems that are driven by an external forces arising from plate tectonic motions. In the study of the relations between seismic dynamics and underlying physical processes, a major role is played by the question of how the magnitude rates evolve with time and if it is at all possible to have a priori estimate of big jumps in the magnitude rate. Recent researchers showed a renewed interest in modeling the earthquake. But still there is an argument whether earthquake is a deterministic or stochastic. OU

Process has an interesting futures in detecting random jumps followed by a mean reverting. As the historical data reveals that earthquake is a stochastic which is not random, here we are going to use one of a financial model to estimate the future major earthquakes. The main objective is to explore and develop mathematical and computation related to estimation of earthquake. The models are complex and new numerical methods need to be devised for solving these problems. We have been working on related subjects like critical phenomena modeling, and more recently we have embarked to work in seismologic events modeling. In earthquake studies, the primary tool which describes the earthquake signal is the ground acceleration signal recorded using a seismometer. Seismometers placed at different locations will record the signal differently, depending on the distance from the epicenter of the earthquake and the soil composition between the epicenter and the location of recording. The earthquake signal exhibited clear stochastic volatility behavior. This led us to believe that the tools we develop in this area will have great importance for the field.

In [34] the authors analyze the signal recorded for the Parkﬁeld county California earthquake of September 28, 2004. The magnitude of this earthquake was 6.0 on the Richter scale. The author analyzed data from two diﬀerent stations: Red Hills and Donna Lee, situated approximately 20 miles apart. In the top left of Figure 6.1 a plot of the raw acceleration signal recorded is provided.

An application of the technique of estimating the variability of the signal using a Markov chain gives the variability (stochastic volatility) estimate in bottom left of Figure 6.1. The two signals

66 are plotted using the same scales and clearly the one recorded further away bears little resemblance with the one recorded closer to the epicenter. An application of the technique of estimating the variability of the signal using a Markov chain gives the variability (stochastic volatility) estimate in theDonna right side Lee/Red of Figure Hills 6.1. Comparisons: Variability Comparison 27 – 47 seconds 27 – 42 Seconds From 17:15 on 9/4/04 Green: Donna Lee Blue: Red Hills

Donna Lee Parkfld CA Sept 4 Accelerometer Readings Accelerometer -100 0 50 150 30 35 40

seconds

R e d H ills P a r k fld C A S e p t 4 Variability Discrete Estimate Variability Discrete 0 5 10 15 20 25 30 Accelerometer Readings Accelerometer -100 0 50 150

30 35 40 30 35 40

seconds seconds

Figure 6.1. Images comparing the raw signal and the variability estimates for two locations. The Red Hills variability is trailing the Donna Lee variability by about 1.3 seconds

A further extension is to use more general L´evyﬂight dynamics for the earthquake signal.

In the study of relations between seismic dynamics and underlying physical processes, a major role is played by the question of how the magnitude rates evolve with time and if it is at all possible to have an a priori estimate of big jumps in the magnitude rate. This is highlighted in the fault interaction model [58] which is based on the hypothesis that small and sudden stress changes cause large changes in seismic rate. Several statistical methods have been developed to serve this purpose. The general framework assumes that the temporal dynamics on a closed time interval of geophysical processes is fully described by the expected numbers of events in the time period. In

[24] the author used Gamma-Ornstein–Uhlenbeck process to capture major earthquakes in certain region of California.

In a recent work [38] scale invariant functions and stochastic L´evymodels are applied to geophysical data and it is shown that a pattern arises from the scale invariance property and

L´evyﬂight models that may be used to estimate parameters related to some major earthquakes.

Modern literature also uses the generalized Omori law and ETAS model (Epidemic-Type aftershock sequence) for quantitative statistic modeling of seismic regime [40, 46, 48]. However, it appears that there is a major drawback with the procedure described in the previous works. Firstly, it is

67 now well accepted that an earthquake model is not a deterministic one and hence the deterministic models such as scale-invariance technique may not be a suitable model.

However, it appears that there is a major drawback with the procedure described in the previous works. Firstly, it is now well accepted that an earthquake model is not a deterministic one and hence the deterministic models such as scale-invariance technique may not be a suitable model. For the existing stochastic models there is no concrete theory that models the data point and estimate the earthquake data from a separate estimation depending on the model parameters.

For example, in [37, 38] earthquake estimation is solely dependent on the data itself and the ﬁrst outlier data gives an estimation. This method is good for some geographical regions but fails signiﬁcantly for most of the other regions. The possible reasons are: (i) those models do not take into account the physical behavior of the earthquake data, and (ii) all the previous models are not completely stochastic.

In this proposal we propose a model of earthquake data based on a completely stochastic process. The model process will depend on some parameters which will be dependent on a particular time frame and a speciﬁc geographical location. One of the most important features of such modeling is the stationary. This means the magnitude process of a geophysical event has invariant statistical distributions for diﬀerent temporal non-overlapping ranges of the same size. A stochastic process (Xt)t≥0 has stationary increments if the law of Xt+h −Xt, with h > 0, does not depend on t. This means that the statistical description of the process over a closed interval of time is invariant with respect to shifts of the starting time provided the same length of time interval is considered.

Stationary is used in literature for modeling diﬀerent geophysical events (see [26, 40, 46, 64] and references therein). For a particular earthquake prone region if the time series of data points

(magnitude) of an earthquake are joined with lines then the following properties are clear for any earthquake data:

1. Magnitude is a non-negative stationary stochastic process.

2. For any ﬁnite interval of time there are only ﬁnite number of jumps.

3. The sample path of magnitude of earthquake consists of upward jumps (signiﬁcant earth-

quake) and gradual decrease (aftershocks).

68 The purpose of this proposal is to use non-Gaussian Ornstein-Uhlenbeck (OU) processes to model the magnitude process of earthquake. Non-Gaussian processes of OU type have considerable potential as building-blocks for different stochastic models of observational time series from a variety of fields. They offer the possibility of capturing important distributional deviations from

Gaussianity and provide ﬂexibility in modeling dependence structure.

Most common Non-Gaussian OU processes in literature are Gamma-OU and Inverse Gaus- sian OU processes. For Inverse Gaussian OU process, it jumps inﬁnitely often in every interval of time and hence it may not be a good candidate for modeling geophysical data. However, Gamma

OU process satisﬁes all the three aforementioned criteria. This proposal aims at developing Gamma-

OU process and its modiﬁcations in relation with the geophysical data, drawing on and extending powerful results from probability theory for applications in stochastic computations. This analysis will eventually lead to the estimation of future major earthquakes.

6.2. Ornstein-Uhlenbeck Processes for Geophysical Modeling

When Lévyprocesses are used as driving noise it is possible to construct a large family of mean-reverting jump processes with linear dynamics on which various properties such as positiveness or the choice of marginal distribution, can be imposed. We consider continuous time stationary and non-negative processes which are defined by the following stochastic differential equation

dMt = −λMt dt + dZλt,M0 > 0, (6.1)

where the process Zt is a subordinator- that is, it is a L´evyprocess with no Gaussian component and positive increments. The rate parameter λ is arbitrary positive and Zt is called the Background

Driving L´evyProcess (BDLP). The unusual timing Zλt is deliberately chosen so that it will turn out that whatever the value of λ the marginal distribution of (Mt) will be unchanged. We call the process M = (Mt)t≥0 to be Ornstein-Uhlenbeck (OU) process. The solution of (6.1) can be explicitly written as

Z t (−λt) Mt = e M0 + exp (−λ(t − s)) dZλs, (6.2) 0

−λt −λt R λt s which can also be written as Mt = e M0 + e 0 e dZs. As Z is an increasing process and

69 M0 > 0, the process (Mt) is strictly positive and it is bounded from below by the deterministic function exp(−λt)M0. Since the process Zt has positive increments and no drift, (Mt) moves up entirely by jumps and then tails off exponentially. Since equation (6.2) is a driven by a subordinator we can apply the definition of (3.0.7) and theorems of (3.0.8) and (3.0.9). We can find the log cumulant function of Mt and Z1 respectively by the identity we saw on chapter 3. It is clear from [51] (Theorem 17.5(ii)) that for any self-decomposable law D there exists a Lévy process Z such that the process of OU type driven by Z has invariant distribution given by D. If

D θD κ (θ) = log E[e ], and κ(θ) is the cumulant transform for Z1, then it is well known (see [3, 6]) that they are related by the fundamental equality

Z ∞ κD(θ) = κ(θ exp(−s))ds. (6.3) 0

This can be expressed as dκD κ(θ) = θ (θ). (6.4) dθ

To prove the above relation let u = θe−s then du = −uds hence the above equation (6.4) can be derived as

Z ∞ κD(θ) = κ(θ exp(−s))ds 0 Z 0 κ(u) = − du, θ u dκD κ(θ) = θ (θ) (6.5) dθ

Lemma 6.2.1. If Mt follows a Γ(ν, α) then its cumulative generating function is given by

α κMt (θ) = ν log (6.6) α − θ

Proof. We refer you to (3.0.12).

70 From the above lemma it is easy to see that the cumulative function of Zt is given by

νθ κ(θ) = . (6.7) α − θ

It has been know that Γ(ν, α) is an infinite divisible which makes it easier to find its Lévy density which is given by u(x) = νx−1e−αx. Using theorem (3.0.9), we can calculate the Lévy density function of the BDLP Z and is

w(x) = ανe−αx. (6.8)

It is clear from (Schoutens [52]) the BDLP for the Γ(ν, α)-OU process is a compound Poisson process N Xt Zt = xn, (6.9) n=1 where (Nt)t≥0 is a Poisson process with E(Nt) = νt and {xn, n = 1, 2,... } is an independent and identically distributed sequence with each xn has Γ(1, α) (that is Exp(α)) law. From the properties of the compound Poisson process it is clear that the stochastic process (Mt)t≥0 has many desired properties such as (see [63]):

• The process Z (which is the BDLP of M) has c´adl´agpiece wise constant functions.

• The jump times of Z (and thus for M) have the same law as the jump times of the Poisson

process Nt.

• Zt has the characteristic function given by

Z ∞ iuZ iux −αx E(e t ) = exp tν (e − 1)αe dx , u ∈ R. (6.10) 0

The stochastic process (Mt) which is defined above BDLP is called the Gamma-OU process. Since the BDLP is compound Poisson process, on any compact time interval it jumps a finite number of times.Immediately this will follow that the Gamma-OU process (Mt) also jumps a finite number of times in every compact time-interval as it is driven by BDLP. Since a selected region of interest have finite number of major earthquakes, We need a model which has a finite number of jumps on

71 a certain compact interval. That is why we choose Gamma-OU process over Inverse Gaussian-OU process as the later jumps inﬁnitely over any compact time interval.

6.3. Computation of Characteristic Function

Assume that the magnitude of the earthquake data is given by M = (Mt)t≥0 (deﬁned in

(6.1) or (6.2)) on some ﬁltered probability space (Ω, F, (F)0≤t≤T ,P ). Estimation of a major event in a future time depends on the analysis of the conditional distribution of the magnitude process

MT given the information up to a certain time t0. In practice t0 is taken to be the time till which the magnitude data for the geophysical process is available and T is taken to be some reasonable time after t0 on which the estimation of future major event is computed. The characteristic function for MT given the information up to time t0 ≤ T is given by E[exp(iuMT )|Ft0 ], where u ∈ R. The conditional distribution of MT given the information up to time t0 ≤ T will be denoted by MT |t0 . The characteristic function of a stochastic process completely characterizes its law. That is, two stochastic processes with the same characteristic function are identically distributed. A characteristic function is always continuous and its value is unity when u = 0.

Additional smoothness properties of the characteristic function depend on the existence of the statistical moments of the stochastic process.

We need the following theorem to attain the main result of this section. We quote this result from [42].

+ Theorem 6.3.1. Let Z be a subordinator with cumulant transform κ and let f : R → C be a complex-valued, left continuous function such that <(f) ≤ 0. Then

Z t Z t E exp f(s) dZλs = exp λ κ(f(s)) ds . (6.11) 0 0

We can now compute the characteristic function of the stochastic process (Mt) given by (6.1) or equivalently by (6.2).

Theorem 6.3.2. Consider the OU-process given by (6.1). Then the characteristic function φ(u) =

72 E[exp(iuMT )|Ft0 ] of MT |t0 is given by

Z T −λT −λ(T −s) φ(u) = exp iuMt0 e + λ κ iue ds , (6.12) t0 where κ(·) is the cumulant transform for Z.

Proof. Substituting equation (6.2) and using the above theorem (6.3.1) we get and < iue−λ(T −s) =

0 we get

φ(u) = E[exp(iuMT )|Ft0 ] Z T

= E exp iu exp(−λT )Mt0 + exp (−λ(T − s)) dZλs |Ft0 t0 Z T −λT −λ(T −s) = exp iue Mt0 E exp iue dZλs t0 Z T −λT −λ(T −s) = exp iuMt0 e + λ κ iue ds . (6.13) t0

It is easy to see that equation (6.12) is true for u = 0. Thus, statistically, all moments of

MT |t0 exist. Also, the probability density of the process MT |t0 can be recovered from a straight- forward generalization of φ(u) to Laplace transform. This is similar to the computations in [42].

We now compute the characteristic function for M when it is Gamma-OU process. Using (6.7) we obtain

Z T Z T iνue−λ(T −s) κ iue−λ(T −s) ds = ds −λ(T −s) t0 t0 α − iue ν = − [2λ(T − t ) + log(α2 + u2) − log α2e2λ(T −t0) + u2 2λ 0 !! α αeλ(T −t0) + 2i arctan − arctan ]. (6.14) u u

Thus for Gamma-OU process substituting (6.14) into theorem (6.12) gives a closed form solution of the characteristic function which is given by

73 ν h i φ(u) = exp[− 2λ(T − t ) + log(α2 + u2) − log α2e2λ(T −t0) + u2 2 0 " !# α αeλ(T −t0) + i uM e−λT − ν arctan + ν arctan ]. (6.15) t0 u u

Thus

h ν h ii |φ(u)| = exp − 2λ(T − t ) + log(α2 + u2) − log α2e2λ(T −t0) + u2 2 0 ν −νλ(T −t ) 2 2 − ν 2 2λ(T −t ) 2 2 = e 0 (α + u ) 2 α e 0 + u ! ν α2 + u2e−2λ(T −t0) 2 = . (6.16) α2 + u2

For the regression analysis in Section 6.5 this theoretical |φ(u)| ∈ R for Gamma-OU processes are ﬁtted with the absolute value of the characteristic function computed from the observed data to estimate the appropriate values of α, ν and λ.

6.4. Analysis of the First-Passage Time

In this section we will give some definition of the exit time that will help us in the earthquake data modeling with a Gamma-OU process. The first passage time or exit time has many interesting applications in different area such as mathematical finance, dam theory reliability analysis, etc.

Here we give the formal deﬁnition stopping time.

Deﬁnition 6.4.1. If U is an open subset of R then the ﬁrst passage time of a stochastic process

(Xt)t≥0 is deﬁned as

τU = inf{t > 0 : Xt ∈/ U}. (6.17)

Thus for a given geophysical data which is modeled by (Mt)t≥0, it is important to approximate the distribution of the first passage time or exit time of a major earthquake. The first passage time has some interesting futures, for example the first passage time is stopping time (or Markov time) with respect to the filtration of the given stochastic process. Also exit time are also used harmonic measure and hitting distribution,for detailed we refer the reader to [22, 47].

If b is a threshold of the magnitude above which earthquakes can be attributed as “major” or “devastating” for a certain geographical region, then it is important to understand the exit time

74 Figure 6.2. Graphical example of exit time. given by

τb = inf{t > 0 : Mt ≥ b}, (6.18)

for some given value b > M0. The distribution of τbusing the transition density of a process is is well understood for the Gaussian OU processes from the pioneering work [22]. However, this problem is signiﬁcantly complicated for non-Gaussian OU processes.

The distribution of the ﬁrst passage time is determined through its Laplace transform which is found by exploiting certain stopped martingales derived from using bounded partial eigen- functions for the inﬁnitesimal generator for the stochastic processes. There are several works in literature which deal with approximation, asymptotic expansions, integral equations, and implicit expressions related to the exit time of non-Gaussian OU processes (see [44, 43, 45, 28, 11]). How-

75 ever, an explicit expression of the distribution of the ﬁrst-passage time for a general non-Gaussian

OU process is still an area of active research interests.

For the analysis of the geophysical data we can reasonably assume

E(log(1 + |Zλ|)) < ∞, (6.19)

which is a necessary and suﬃcient condition of convergence of (Mt) in distribution to a proper limit

PNλt (see [67]). When M is a Gamma-OU process, clearly Zλt = n=1 xn, is a compund poisson process

, where E(Nλ) = νλ and thus the moment generating function for Zλ can be obtained from (6.10) as Z ∞ E(evZλ ) = exp λν (evx − 1)αe−αx dx . (6.20) 0

As used in [11], set

K = sup{u ≥ 0 : E(euZλ ) < ∞}, (6.21) and 1 Z u log(E(evZλ )) Φ(u) = dv, 0 ≤ u < K. (6.22) λ 0 v

We need the following simple Lemma to have an estimate of the expectation of the ﬁrst passage time for Gamma-OU processes.

Lemma 6.4.2. When M is a Gamma-OU process K = α and

α Φ(u) = ν log , 0 ≤ u < α. (6.23) α − u

Proof. It follows directly from (6.20) that K = α. Moreover, simple calculation shows

Z ∞ log(E(evZλ )) = λν (evx − 1)αe−αx dx 0 ! e(v−α)x e−αx = λνα + |∞ v − α α x=0 λνv = . (6.24) α − v

α The last statement comes from the fact that v < α. Thus for 0 ≤ u < α, Φ(u) = ν log α−u .

76 It is clear from Lemma 6.4.2 that when M is a Gamma-OU process Φ(K) = Φ(α) = +∞.

R α uz−Φ(u) µ−1 Also, for this case, it is evident from [43] that if G(z, µ) = 0 e u du, with µ > 0 and Φ(u) deﬁned as in (6.23), then

−µλτb E(e G(Mτb , µ)) = G(Mt0 , µ), µ > 0. (6.25)

We now state the main result for estimation of the ﬁrst passage time of the process M.

Theorem 6.4.3. Let M be a Γ(ν, α)-OU process given by (6.1). Then the expected value of τb satisﬁes the following relation

ν 1 Z α (eub − euM0 ) α − u E(τb) ≥ du. (6.26) λ 0 u α

Proof. We observe that (6.19) is satisﬁed and 0 < K = α < ∞ and Φ(K) = +∞. Thus all conditions of [11] (Theorem 2) are satisﬁed. Hence

Z K uMτ uM0 1 (e b − e ) −Φ(u) E(τb) = E e du λ 0 u 1 Z α (eub − euM0 ) ≥ e−Φ(u) du λ 0 u ν 1 Z α (eub − euM0 ) α − u = du, (6.27) λ 0 u α where in the last step result for Φ(u) from Lemma 6.4.2 is used

Theorem 6.4.3 plays a key role in Section 6.5. The parameters α, ν and λ will be found using regression analysis. Thus for the Gamma-OU model we can estimate the expected value of the major events using (6.26). This gives the estimated time after which the awareness level should be raised signiﬁcantly that a major event will follow.

6.5. Statistical Analysis of the Data

In this section we analyzed the earthquake data using method of non-linear least square

(nls) estimation regression. This nls use the given data and try and ﬁnds the given parameter which minimzes the least square error. We used a statistical software which is called R to do the

77 analysis and get the root mean square error (RMSE). the ﬁrst part of this section (Section 6.5.1) we describe the geophysical data set used for the computation. In Section 6.5.2 we give a detailed account of our results with comparison with existing results.

6.5.1. Geophysical (Earthquake) Data

The geophysical data was obtained from U.S. Geological Survey (USGS) from January 1,

1973 to November 9, 2010 for a certain region in California. This data contains information about the date, longitude, latitude, and magnitude of each recorded earthquake in the region.

This study will be characterized by the geographical region in a neighborhood of location of the major earthquake. The choice of this region should be done carefully. This area can not be too small because in that case there will be not enough data to run the regression analysis. The area can not be too big because in that case the magnitude data set is distorted due to noise from unrelated events. As in [38] the data is obtained using a square centered at the coordinates of the major event. The sides of the square are usually chosen as ±0.1◦–0.5◦ in latitude and ±0.2◦–0.5◦ in longitude. A segment 0.1◦ of latitude at the equator is ≈ 6.9 miles ≈ 11.11 km in length.

The earthquake magnitude is the recorded data used for 5 diﬀerent square regions and the model is ﬁtted to get the parameter estimate. The policy of the USGS regarding recorded magnitude is the following [1]: (i) Magnitude is a dimensionless number between 1 and 12. (ii) The reported magnitude should be moment magnitude, if available. (iii) The least complicated, and probably most accurate, terminology is to just use the term “magnitude” and to use the symbol

M for it.

The magnitude is recorded in the data used and where available moment magnitude is used.

A major earthquake event is deﬁned as an earthquake with magnitude greater than 7. For more information we refer to the speciﬁc USGS documentation available at http://earthquake.usgs.gov/aboutus/docs/020204mag policy.php.

6.5.2. Regression Analysis and Estimation of Major Events

For the regression analysis we use data prior to a major earthquake event. The time series is shifted in such a way that the initial time is 0 and the ﬁnal time (up to which data is available to estimate future major event) is t0. For the model t0 is taken to be the time before a major earthquake for which the magnitude is actually recorded. Consequently, for a particular geographical location, M0 and Mt0 denote the magnitude of the data for the recorded initial and

78 final times respectively. The time T can be taken to be any day between t0 and the day of the major earthquake. Choice of T is on users’ discretion as long as the time series for the magnitude process of earthquake in a geographical region is provided up to time t0 < T . However, the two important aspects to keep in mind are: (i) T can not be too large. The geophysical data is not provided for small magnitudes which potentially distort the probability distribution of the magnitude process from the data. If too long is waited after t0 it is very much possible that the probability distribution from the computed data set is already changed quite significantly. (ii) T can not be too small. Because, in that case there may not be any jump in the Lévyprocess Z and thus the contribution from the stochastic term in (6.12) is zero (or close to zero). So the model itself becomes insignificant.

Hence it is important to choose T carefully. For example, in our regression analysis T is taken to be t0 + 1.

Based on the data of the magnitude process (Mt)0≤t≤t0 it is possible to compute the characteristic function φO(u) = E[exp(iuMT )|Ft0 ], where the suffix ‘O’ of φ stands for “observed” characteristic function based on the data set up to time t0. We numerically compute the probability distributions and then use Fast Fourier Transform (FFT) to compute φO(u). We take modulus of these complex function to get |φO(u)|, which is a real function of u. For the Gamma-OU model, we fit (6.16) to the observed |φO(u)| using non-linear regression. The fitting minimizes root-mean square error (RMSE). The software which is used for regression is R. This gives the best fit values of the parameters α, ν and λ. We find for each specific region, p-value is much less than 0.05 for all the parameters which shows that all the three parameters (α, ν and λ) are very significant for the model. Finally (6.26) is used to compute the lower bound of the expected time of a major earthquake. At that time the awareness level should be raised significantly in anticipation of a major earthquake. We note that (6.26) does not give a lower bound for τb. Instead it gives the lower bound for E(τb) which is a statistical property of τb. We acknowledge the fact that in some exceptional cases it is still possible to get a major earthquake even before the lower bound time of

E(τb). However, statistically those incidences are very less probable. Partial compensation of the lower bound of E(τb) can be attained by little ﬂexibility in the values of b in the deﬁnition of major earthquake. For our calculation b is chosen from a range 6.2 to 6.7.

Table 6.1 presents the estimated parameters from regression analysis. Table 6.2 presents the

79 estimations from the regression model. In Table 6.2, tr is the estimated date on which alert should be raised about possible earthquake in near future, and tm is the actual known major earthquake date.

The following Figures (Figure 6.3 to Figure 6.7) present the fitting of absolute value of the characteristic function. In each of the figures, circles denote values of |φO(u)| which are computed from actual data points and the solid line is the regression fitting.

It is clear from Table 6.2 that the Gamma-OU model can be very effective in modeling magnitude process of earthquake in certain geographical locations. At the time tr a near-future earthquake warning should be issued. Observe that tr is always less than tm which is why this model is reasonable for estimating earthquake date. For the previous works where the Ising model, scale invariance or Lévyflights are used [30, 39, 37, 38] the estimated date is typically but not always preceding the major event. For different deterministic models the estimation is based on finding some accumulation point of a sequence of numbers. However, there is no reason that the limit point is before the actual date of major earthquake. As shown in the following tables and figures,

Gamma-OU works surprisingly good in that respect. On the other hand, for the stochastic L´evy

flight models in literature, the estimated date was found from the given data set and some points in that data set may be after a major earthquake. For example, for the Lévy flight model with unit variance the estimation is based on the deviation of the data set from the cumulative distribution of the model [38]. However, that deviation can very well occur after the major earthquake. The model proposed in this study do not take into account the data after major earthquake. Gamma-

OU model only takes the historical data and thus more realistic for practical applications.

6.6. Conclusion

In this chapter we use Gamma-OU process to model magnitude process of earthquake data.

The predicted time is typically preceding the time of the major event. This methodology can be used in real time by looking at the minor earthquakes up to a certain date to estimate a major earthquake in future. However, there are some diﬃculties with this method in terms of false alarm.

When this model is used to analyze some other data sets most of the time it works reasonably.

However, sometime the predicted time is within some days after the ﬁrst data. That is of course correct (in the sense that the estimated time precedes the actual time) but sort of false alarm to the situation as no major earthquake was reported near to that point. There are two ways two

80 improve those situations. Firstly, we can modify the value of the major earthquake b so that tr becomes bigger than t0. Such tr (with reasonable b) will be the new estimation. Secondly, we can modify the Gamma-OU model to a superimposition of several OU processes. This will allow the model to be the sum, or superimposition, of independent OU processes. As the processes do not need to be identically distributed, this offer some extra flexibility in the model. Modified processes are potential candidates for modeling long-range dependence and self-similarity (see for example

[3, 6] and references therein). Additional work is in progress on this aspect in order to improve the present model.

81 ●

0.8 ●

● 0.6

● 0.4

●

● Abs. value of characteristic function value Abs. ● ● 0.2 ● ● ● ● ● ● ● ● ● ●

5 10 15

Figure 6.3. Fitting for Latitude 40.37◦ ± 0.1◦ and Longitude −124.32◦ ± 0.2◦.

● 0.8 0.6 ● 0.4

●

● ● ●

0.2 ● ● ● ● Abs. value of characteristic function value Abs. ● ● ● ● ●

● ● 0.0

5 10 15

Figure 6.4. Fitting for Latitude −23.34◦ ± 0.5◦ and Longitude −70.30◦ ± 0.5◦.

82 ● 0.8

●

0.6 ● modes ● 0.4

●

● ● ● 0.2

● ● ● ● ● ● ●

2 4 6 8 10 12 14

Figure 6.5. Fitting for Latitude 63.52◦ ± 0.17◦ and Longitude −147.44◦ ± 0.34◦.

● 0.8

● 0.6

● 0.4

● Abs. value of characteristic function value Abs. 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

5 10 15 20

Figure 6.6. Fitting for Latitude −17.66◦ ± 0.03◦ and Longitude −178.76◦ ± 0.06◦.

83 ● 0.8

● 0.6

● 0.4

● Abs. value of characteristic function value Abs. 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

5 10 15 20

Figure 6.7. Fitting for Latitude −19.93◦ ± 0.05◦ and Longitude −178.18◦ ± 0.1◦.

Table 6.1. Results from regression analysis

Latitude Longitude M0 t0 α ν λ RMSE 40.37◦ ± 0.1◦ −124.32◦ ± 0.2◦ 3.9 12/20/1991 2.4113 1.2045 2.3879 0.06346 −23.34◦ ± 0.5◦ −70.30◦ ± 0.5◦ 4.6 06/01/1995 2.0007 0.4244 0.9951 0.05913 63.52◦ ± 0.17◦ −147.44◦ ± 0.34◦ 4.2 11/03/2002 2.8355 2.3713 0.99289 0.06875 −17.66◦ ± 0.03◦ −178.76◦ ± 0.06◦ 4.1 07/15/2004 2.7123 2.7231 0.99052 0.04582 −19.93◦ ± 0.05◦ −178.18◦ ± 0.1◦ 4.7 08/25/2005 3.2601 1.6960 0.99552 0.03451

Table 6.2. Estimation from the regression model.

Latitude Longitude t = 0 tr tm 40.37◦ ± 0.1◦ −124.32◦ ± 0.2◦ 09/30/1973 6572 6782 −23.34◦ ± 0.5◦ −70.30◦ ± 0.5◦ 07/13/1973 7759 8052 63.52◦ ± 0.17◦ −147.44◦ ± 0.34◦ 03/26/1976 10632 10753 −17.66◦ ± 0.03◦ −178.76◦ ± 0.06◦ 01/17/1973 11394 11501 −19.93◦ ± 0.05◦ −178.18◦ ± 0.1◦ 04/04/1976 10454 10865

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